20082009 School Handbook: Volume I

    Mathcounts target practice questions. May 7, 2009 the 20072008 mathcounts question writing committee developed the school handbook and mathcounts competition problems from the pastprepare students for the target and team rounds of competition. engineering.utep.edu.

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© 2008 MATHCOUNTS Foundation
1420 King Street, Alexandria, VA 22314
703-299-9006 [email protected]
www.mathcounts.org
Unauthorized reproduction of the contents of this publication is a violation of applicable laws.
Materials may be duplicated for use by U.S. schools.
MATHCOUNTS
®
and Mathlete
®
are registered trademarks of the MATHCOUNTS Foundation.
2008–2009
School Handbook:
Volume I
For questions about your local MATHCOUNTS program,
please contact your local (chapter) coordinator. Coordinator contact information
is available in the Find a Coordinator section of www.mathcounts.org.
Contains 100 creative math problems
that meet NCTM standards for grades 6-8.
The printing of this handbook, accompanying registration materials
and their distribution was made possible by
Acknowledgments
The 2007–2008 MATHCOUNTS Question Writing Committee developed the questions for the
2008–2009 MATHCOUNTS School Handbook and competitions:
William Aldridge, Springî‚¿eld, Va.
Mady Bauer, Bethel Park, Pa.
Susanna Brosonski, Orlando, Fla.
Lars Christensen (STE 89), Minneapolis, Minn.
Dan Cory (NAT 84, 85), Seattle, Wash.
Craig Countryman, San Diego, Calif.
Roslyn Denny, Valencia, Calif.
Edward Early (STE 92), Austin, Texas
Nancy English, Glendale, Mo.
Barry Friedman (NAT 86), Scotch Plains, N.J.
Joan M. Gell, Redondo Beach, Calif.
Dennis Hass, Westford, Mass.
Bonnie Hayman, St. Louis, Mo.
Helga Huntley (STE 91), Seattle, Wash.
National Judges review competition materials, develop Masters Round questions and
serve as arbiters at the National Competition:
National Reviewers proofread and edit the MATHCOUNTS School Handbook and/or competition materials:
Doug Keegan (STE 91, NAT 92), Austin, Texas
David Kung (STE 85, NAT 86), St. Mary’s City, Md.
Jane Lataille, Los Alamos, N.M.
Stanley Levinson, P.E., Lynchburg, Va.
Artie McDonald, P.E. (STE 88), Melbourne, Fla.
Paul McNally, Haddon Heights, N.J.
Randy Rogers, Cedar Rapids, Iowa
Frank Salinas, Houston, Texas
Laura Taalman (STE 87), Harrisonburg, Va.
Craig Volden (NAT 84), Columbus, Ohio
Chaohua Wang, Bloomington, Ill.
Deborah Wells, Rockville, Md.
Judy White, Littleton, Mass.
Yiming Yao (STE 96), Vancouver, British Columbia
•
Chair: Sandy Powers, College of Charleston, Charleston, S.C.
• Sam Baethge, San Antonio, Texas
• Chengde Feng, Oklahoma School of Science and Mathematics, Oklahoma City, Okla.
• Joyce Glatzer, West New York Public Schools, West Patterson, N.J.
• Greg Murray, Dixie High School, St. George, Utah
• Tom Price, Norris Middle School, Firth, Neb.
• Susan Wildstrom, Walt Whitman High School, Bethesda, Md.
• Richard Case, Computer Consultant, Greenwich, Conn.
• Flavia Colonna, George Mason University, Fairfax, Va.
•
Peter Kohn, James Madison University, Harrisonburg, Va.
• Carter Lyons, James Madison University, Harrisonburg, Va.
• Monica Neagoy, Mathematics Consultant, Washington, D.C.
• Dave Sundin (STE 84), Statistics and Logistics Consultant, San Mateo, Calif.
Editor and Contributing Author: Jessica Liebsch, Manager of Education
MATHCOUNTS Foundation
Introduction and Content Editor: Kristen Chandler, Deputy Director & Director of Education
MATHCOUNTS Foundation
Executive Director: Louis DiGioia
MATHCOUNTS Foundation
Honorary Chair: William H. Swanson
Chairman and CEO
Raytheon Company
The Solutions were written by Kent Findell, Diamond Middle School, Lexington, Mass.
MathType software for handbook development contributed by Design Science Inc., www.dessci.com, Long Beach, Calif.
The National Association of Secondary School
Principals has placed this program on the NASSP
Advisory List of National Contests and Activities
for 2008–2009.
Count Me In!
A contribution to the
MATHCOUNTS Foundation
will help us continue to make
this worthwhile program
available to middle school students
nationwide.
The MATHCOUNTS Foundation
will use your contribution for
programwide support to give
thousands of students the
opportunity to participate.
To become a partner in
MATHCOUNTS, send
your contribution to:
MATHCOUNTS Foundation
P.O. Box 1338
Merriî‚¿eld, VA 22116-9706
Or give online at:
www.mathcounts.org
Other ways to give:
• Ask your employer about
matching gifts. Your donation
could double.
• Remember MATHCOUNTS
in your United Way and
Combined Federal Campaign
at work.
• Leave a legacy. Include
MATHCOUNTS in your will.
For more information regarding
contributions, call the director
of development at 703-299-9006,
ext. 103 or e-mail
[email protected]
The MATHCOUNTS Foundation is a 501(c)3
organization. Your gift is fully tax deductible.
The American Society of Association Executives
has recognized MATHCOUNTS with a 2001 Award
of Excellence for its innovative, society-enriching
activities.
TABLE OF CONTENTS
Critical 2008–2009 Dates ............................................................. 4
Introduction
.................................................................................. 5
New This Year
............................................................................... 5
The MATHCOUNTS Web Site.......................... 5
The MATHCOUNTS OPLET............................ 7
Building a Competition and/or Club Program ......................... 8
Recruiting Mathletes
®
....................................... 8
Maintaining a Strong Program ........................... 8
MATHCOUNTS Competition Program .................................... 9
Preparation Materials ......................................... 9
Coaching Students ........................................... 10
Ofî‚¿cial Rules & Procedures ........................... 12
Registration ................................................. 12
Eligible Participants .................................... 13
Levels of Competition ................................ 14
Competition Components ........................... 15
Additional Rules ......................................... 16
Scoring ....................................................... 17
Results Distribution .................................... 17
Forms of Answers ....................................... 18
MATHCOUNTS Club Program
............................................... 19
Club Materials .................................................. 19
Rules, Procedures & Deadlines ....................... 19
Getting Started ............................................ 19
Attaining Silver Level Status ...................... 20
Attaining Gold Level Status ........................ 20
Frequently Asked Questions ............................ 21
Handbook Problems
Warm-Ups and Workouts ................................. 23
Stretch .............................................................. 32
Resources
Problem-Solving Strategies ............................................ 33
Vocabulary and Formulas
............................................... 45
References
....................................................................... 47
Answers to Handbook Problems
.............................................. 49
Solutions to Handbook Problems
............................................. 53
Problem Index
............................................................................ 59
Request/Registration Form
....................................................... 63
The MATHCOUNTS Foundation makes its products and services available on a non-discriminatory basis. MATHCOUNTS does not discriminate
on the basis of race, religion, color, creed, gender, physical disability or ethnic origin.
 MATHCOUNTS 2008–2009
CRITICAL 2008–2009 DATES
Immediately For easy reference, write your local coordinator’s address and phone number here.
Contact information for coordinators is available in the Find a Coordinator section of
www.mathcounts.org or from the national ofî‚¿ce.
 September- Send in your school’s Request/Registration Form to receive Volume II of the handbook,
Dec. 12 the Club in a Box resource kit and/or your copy of the 2008–2009 School
Competition. Items will ship shortly after receipt of your form, with mailing of
the School Competition Kit following this schedule:
Registration forms postmarked by Oct. 1: Kits mailed in early November.
Kits continue mailing every two weeks.
Registration forms postmarked by Dec. 12 deadline: Kits mailed in early January.
Mail or fax the MATHCOUNTS Request/Registration Form (with payment if
participating in the competition) to:
MATHCOUNTS Registration, P.O. Box 441, Annapolis Junction, MD 20701
Fax: 301-206-9789 (Please fax or mail, but do not do both.)
Questions? Call 301-498-6141 or conî‚¿rm your registration via the Registered Schools
database and/or 2008–2009 Bronze Level Schools list at www.mathcounts.org in the
Registered Schools or Club Program sections, respectively.

Dec. 12 Competition Registration Deadline
In some circumstances, late registrations may be accepted at the discretion of
MATHCOUNTS and the local coordinator. Register on time to ensure participation
by your students.

Mid-January If you have not been contacted with details about your upcoming competition, call
your local or state coordinator!
If you have not received your School Competition Kit by the end of January, contact
MATHCOUNTS at 703-299-9006.

Feb. 1–28 Chapter Competitions
March 1–28 State Competitions
March 6 Deadline for Math Clubs to reach MATHCOUNTS Silver Level & entry into drawing
March 27 Deadline for Math Clubs to reach MATHCOUNTS Gold Level & entry into drawing
May 7–10 2009 Raytheon MATHCOUNTS National Competition at the Walt Disney World
Swan and Dolphin Resort

Interested in more coaching materials or MATHCOUNTS items?
Additional FREE resources are available at www.mathcounts.org.
Purchase items from the MATHCOUNTS store at www.mathcounts.org or contact Sports Awards at 800-621-5803.
Select items also are available at www.artofproblemsolving.com.
(postmarked by)
(received by)
(received by)
MATHCOUNTS 2008–2009 
INTrODUCTION
The mission of MATHCOUNTS is to increase enthusiasm for and enhance achievement in middle school
mathematics throughout the United States. Currently in our 26th year, MATHCOUNTS meets its mission
by providing two separate, but complementary, programs for middle school students: the Competition
Program and the MATHCOUNTS Club Program.
The MATHCOUNTS Competition Program is a program to excite and challenge middle
school students. With four levels of competition — school, chapter (local), state and
national — the Competition Program provides students with the incentive to prepare
throughout the school year to represent their schools at these MATHCOUNTS-hosted*
events. MATHCOUNTS provides the preparation materials and the competition
materials, and with the leadership of the National Society of Professional Engineers,
more than 500 Chapter Competitions, 57 State Competitions and one National
Competition are hosted each year. These competitions provide students with the opportunity to go head-
to-head against their peers from other schools, cities and states; to earn great prizes individually and
as part of their school team; and to progress through to the 2009 Raytheon MATHCOUNTS National
Competition at the Walt Disney World Swan and Dolphin Resort. There is a registration fee for students
to participate in the Competition Program, and participation past the School Competition level is limited
to the top eight students per school. A more detailed explanation of the Competition Program follows on
pages 9 through 18.
The MATHCOUNTS Club Program (MCP) is designed to increase enthusiasm for
math by encouraging the formation within schools of math clubs that conduct fun
meetings using a variety of math activities. The activities provided for the MCP
foster a social atmosphere, and there is a focus on students working together as a
club to earn recognition and rewards in the MATHCOUNTS Club Program. Some
rewards are participation based, while others are achievement based, but all rewards
require a minimum of 12 club members. Therefore, there is an emphasis on building a strong club and
encouraging more than just the top math students within a school to join. There is no cost to sign up for
the MATHCOUNTS Club Program, but a Request/Registration Form must be submitted to receive the
club materials. (A school that registers for the Competition Program is automatically signed up for the
MATHCOUNTS Club Program.) A more detailed explanation of the MATHCOUNTS Club Program
follows on pages 19 through 22.
NEw THIS YEAr
THE MATHCOUNTS wEB SITE
... a new look and great new features designed especially for coaches
The general public still has access to many great materials and resources on www.mathcounts.org,
including the Problem of the Week, Volume I of the 2008–2009 MATHCOUNTS School Handbook,
general information about the MATHCOUNTS Competition Program and Club Program and archived
materials. However, a new Coaches section of the site will provide members-only access to restricted
pages with fantastic resources and networking opportunities.
MATHCOUNTS is able to grant coaches access to portions of the web site that are restricted from the
general public once coaches have completed two simple steps:
1. Create a User Proî‚¿le on mathcounts.org.
2. Either sign up a club with the MATHCOUNTS Club Program or register students for the
Competition Program.
*While MATHCOUNTS provides the actual School Competition Booklet with the questions, answers and procedures necessary to run the School
Competition, the administration of the School Competition is up to the MATHCOUNTS coach in the school. The School Competition is not
required; selection of team and individual competitors for the Chapter Competition is entirely at the discretion of the school coach and need not
be based solely on the School Competition scores.
6 MATHCOUNTS 2008–2009
Once MATHCOUNTS links your User Proî‚¿le with a Club or Competition school, you will have special
resources available to you each time you log in to mathcounts.org.
To create your User Prole:
1. Go to www.mathcounts.org.
2. Click on the purple Login button at the top,
right-hand side of the page (see screen shot to
the left).
3. Click on the New User Proî‚¿le link at the
bottom of the form (see screen shot below).
4. Complete the online form.
The Coaches Home Page
Shortly after your New User Proî‚¿le request
is received, MATHCOUNTS will verify your
coach status and allow you to join the online
community. (Note: You must (1) have a User
Proî‚¿le and (2) send in your Request/Registration
Form for the MATHCOUNTS Club Program
and/or Competition Program for us to grant you
this access. The order in which these are done is
not important, but both must be done to gain the
extra access.) At that time, you will be able to
log in and view your new MATHCOUNTS home
page. Among other items, there will be a Coaches
button in the left-hand navigation menu and an
MCP Members Only link in the Club Program
section (see screen shot to the left).
MATHCOUNTS 2008–2009 7
THE MATHCOUNTS OPLET
(Online Problem Library and Extraction Tool)
... a database of 10,000 MATHCOUNTS problems and the ability to generate
worksheets, ash cards and Problems of the Day
Through www.mathcounts.org, MATHCOUNTS is offering the MATHCOUNTS OPLET — a database
of 10,000 problems and the ability to create personalized worksheets, ash cards and Problems of the Day
with these problems. After purchasing a 12-month subscription to this online resource, the user will have
access to MATHCOUNTS School Handbook
and MATHCOUNTS competition problems from the past
10 years and the ability to extract the problems in personalized formats. (Each format is generated into a
pdf to be printed.)
Imagine the time that can be saved preparing for club meetings,
practice sessions or classroom teaching!
Once the subscription is purchased, the user can access the MATHCOUNTS OPLET each time s/he goes
to www.mathcounts.org and logs in. Once on the MATHCOUNTS OPLET page, the user can tailor the
output to his/her needs by answering a few questions. Among the options that can be personalized are:
Format of the output: Worksheet, Flash cards or Problems of the Day
Number of questions to include
Math concept: Arithmetic, Algebra, Geometry, Counting and Probability, Number Theory, Other
or Random Sampling
MATHCOUNTS usage: Problems without calculator usage (Sprint Round/Warm-Up), Problems
with calculator usage (Target Round/Workout/Stretch), Team problems with calculator usage
(Team Round), Quick problems without calculator usage (Countdown Round) or Random
Sampling
Difî‚¿culty Level: Easy, Easy/Medium, Medium, Medium/Difî‚¿cult, Difî‚¿cult or Random Sampling
Year range from which problem was originally used in MATHCOUNTS materials: Problems are
grouped in î‚¿ve-year blocks in the system
Once these criteria have been selected, the user either (1) can opt to have the computer select the
problems at random from an appropriate pool of problems or (2) can select the problems from this
appropriate pool of problems him/herself.
How does a person gain access to this incredible resource as soon as possible?
A 12-month subscription to the MATHCOUNTS OPLET can be purchased at www.mathcounts.org. The
cost of a subscription is $275; however, schools registering students in the MATHCOUNTS Competition
Program will receive a $5 discount per registered student.
Once MATHCOUNTS processes your subscription payment for the MATHCOUNTS OPLET, you will
be given access to the MATHCOUNTS OPLET page and sent notiî‚¿cation that you are a registered user.
Note: You î‚¿rst must create a User Proî‚¿le on www.mathcounts.org in order for us to grant you access to
this resource.
If you would like to get a sneak peek at this invaluable resource before making your purchase, you can
check out screen shots of the MATHCOUNTS OPLET at www.mathcounts.org. You will see the ease
with which you can create countless materials for your Mathletes, club members and classroom students.
♦
♦
♦
♦
♦
♦
8 MATHCOUNTS 2008–2009
BUILDINg A COMPETITION AND/Or
CLUB PrOgrAM
rECrUITINg MATHLETES
®
Ideally, the materials in this handbook will be incorporated into the regular classroom curriculum so
that all students learn problem-solving techniques and develop critical thinking skills. When a school’s
MATHCOUNTS Competition Program and/or Club Program is limited to extracurricular sessions, all
interested students should be invited to participate regardless of their academic standing. Because the
greatest beneî‚¿ts of the MATHCOUNTS program are realized at the school level, the more Mathletes
involved, the better. Students should view their experience with MATHCOUNTS as fun, as well as
challenging, so let them know from the very î‚¿rst meeting that the goal is to have a good time while
learning.
Some suggestions from successful competition and club coaches on how to stimulate interest at the beginning
of the school year:
• Build a display case using MATHCOUNTS shirts and posters. Include trophies and photos from
previous years’ club sessions or competitions.
• Post intriguing math questions (involving specic school activities and situations) in hallways, the
library and the cafeteria. Refer students to the î‚¿rst meeting for answers.
• Make a presentation at the rst pep rally or student assembly.
• Approach students through other extracurricular clubs (e.g., science club, computer club, chess club).
• Inform parents of the benets of MATHCOUNTS participation via the school newsletter or PTA.
• Create a MATHCOUNTS display for Back-to-School Night.
• Have former Mathletes speak to students about the rewards of the program.
• Incorporate the Problem of the Week from the MATHCOUNTS web site (www.mathcounts.org) into
the weekly class schedule.
• Organize a MATHCOUNTS Math Club.
MAINTAININg A STrONg PrOgrAM
Keep the school program strong by soliciting local support and focusing attention on the rewards of
MATHCOUNTS. Publicize success stories. Let the rest of the student body see how much fun Mathletes
have. Remember, the more this year’s students get from the experience, the easier recruiting will be next
year. Here are some suggestions:
• Publicize MATHCOUNTS meetings and events in the school newspaper and local media.
• Inform parents of meetings and events through the PTA, open houses and the school newsletter.
• Schedule a special pep rally for the Mathletes.
• Recognize the achievements of club members at a school awards program.
• Have a students versus teachers Countdown Round and invite the student body to watch.
• Solicit donations from local businesses to be used as prizes in practice competitions.
• Plan retreats or eld trips for the Mathletes to area college campuses or hold an annual reunion.
• Take photos at club meetings, coaching sessions and/or competitions and keep a scrapbook.
• Distribute MATHCOUNTS shirts to participating students.
• Start a MATHCOUNTS summer school program.
• Encourage teachers of lower grades to participate in mathematics enrichment programs.
MATHCOUNTS 2008–2009 9
The MATHCOUNTS Foundation administers this math enrichment,
coaching and competition program with a grassroots network
of more than 17,000 volunteers who organize MATHCOUNTS
competitions nationwide. Each year more than 500 local competitions
and 57 “state” competitions are conducted, primarily by chapter and state societies of the National Society
of Professional Engineers. All 50 states, the District of Columbia, Puerto Rico, Guam, Virgin Islands,
Northern Mariana Islands, and U.S. Department of Defense and U.S. State Department schools worldwide
participate in MATHCOUNTS. Here’s everything you need to know to get involved...
PrEPArATION MATErIALS
The annual MATHCOUNTS School Handbook provides the basis for coaches and volunteers to coach
student Mathletes
on problem-solving and mathematical skills. Coaches are encouraged to make maximum
use of MATHCOUNTS materials by incorporating them into their classrooms or by using them with
extracurricular math clubs. Coaches also are encouraged to share this material with other teachers at their
schools as well as with parents.
The 2008–2009 MATHCOUNTS School Handbook is in two volumes. Volume I contains 100 math
problems, and Volume II contains 200 math problems. As always, these 300 FREE, challenging and
creative problems have been written to meet the National Council of Teachers of Mathematics’ Standards
for grades 6-8. Volume I is being sent directly to every U.S. school with 7th- and/or 8th-grade students
and any other school that registered for the MATHCOUNTS Competition Program last year. This volume
also is available for schools with 6th-grade students. Volume II of the handbook also will be provided
to schools free of charge. However, Volume II will be sent only to those coaches who sign up for the
MATHCOUNTS Club Program or register for the MATHCOUNTS Competition Program.
In addition to the 300 great math problems, be sure to take advantage of the following resources that are
included in the 2008-2009 MATHCOUNTS School Handbook:
Problem-Solving Strategies
are explained on pages 33-43. Answers to all problems in the
handbook include one-letter codes indicating possible, appropriate problem-solving
strategies.
Vocabulary and Formulas are listed on pages 45-46.
A Problem Index is provided on page 59 to assist you in incorporating the
MATHCOUNTS
School Handbook problems into your curriculum. This index includes problems from
Volumes I and II of the handbook and organizes the problems by topic.
A variety of additional information and resources are available on the MATHCOUNTS web site at
www.mathcounts.org, including problems and answers from the previous year’s Chapter and State
Competitions, the MATHCOUNTS Coaching Kit, MATHCOUNTS Club Program resources, forums and
links to state programs. When you sign up for the Club Program or Competition Program (and you have
created a User Proî‚¿le on the site), you will receive access to even more free resources that are not visible
nor available to the general public. Be sure to create a User Proî‚¿le as soon as possible and then visit the
Coaches section of the site.
NEW this year is the MATHCOUNTS OPLET, which contains MATHCOUNTS School Handbook and
competition problems from the last 10 years. Once a 12-month subscription is purchased, the user can
create customized worksheets, ash cards and Problems of the Day using this database of questions.
For more information, see page 7 or go to www.mathcounts.org and check out some screen shots of the
MATHCOUNTS OPLET. A 12-month subscription can be purchased online at www.mathcounts.org.
MATHCOUNTS COMPETITION
PrOgrAM...
A MOrE DETAILED LOOk
10 MATHCOUNTS 2008–2009
Additional coaching materials and novelty items may be ordered through Sports
Awards. An order form, with information on the full range of
products, is available in the MATHCOUNTS Store section of
www.mathcounts.org or by calling Sports Awards toll-free at
800-621-5803. Interested in placing an online order? A limited
selection of MATHCOUNTS materials is also available at
www.artofproblemsolving.com.
COACHINg STUDENTS
The coaching season begins at the start of the school year. The sooner you begin your coaching sessions,
the more likely students will still have room in their schedules for your meetings and the more preparation
they can receive before the competitions.
Though Volume I has plenty of problems to get your students off to a great start, be sure to request your
copy of Volume II of the 2008–2009 MATHCOUNTS School Handbook as soon as possible (see page
64). Do not hold up the mailing of your Volume II because you are waiting for a purchase order
to be processed or a check to be cut by your school for the registration fee. Fill out your Request/
Registration Form and send in a photocopy of it without payment. We immediately will mail your
Club in a Box resource kit (which contains Volume II of the MATHCOUNTS School Handbook) and
credit your account once your payment is received.
The 300 original problems found in Volumes I and II of the MATHCOUNTS School Handbook are
divided into three sections: Warm-Ups, Workouts and Stretches. Each Warm-Up and Workout contains
problems that generally survey the grades 6-8 mathematics curricula. Workouts assume the use of a
calculator; Warm-Ups do not. The Stretches are collections of problems centered around a speciî‚¿c topic.
The problems are designed to provide Mathletes with a large variety of challenges and prepare them
for the MATHCOUNTS competitions. (These materials also may be used as the basis for an exciting
extracurricular mathematics club or may simply supplement the normal middle school mathematics
curriculum.)
Answers to all problems in the handbook include one-letter codes indicating possible, appropriate
problem-solving strategies. These strategies are explained on pages 33-43.
wArM-UPS AND wOrkOUTS
The Warm-Ups and Workouts are on pages 23-31 and are designed to increase in difî‚¿culty as students go
through the handbook.
For use in the classroom, the problems in the Warm-Ups and Workouts serve as excellent additional
practice for the mathematics that students are already learning. In preparation for competition, the
Warm-Ups can be used to prepare students for problems they will encounter in the Sprint Round. It is
assumed students will not be using calculators for Warm-Up problems. The Workouts can be used to
prepare students for the Target and Team Rounds of competition. It is assumed students will be using
calculators for Workout problems. Along with discussion and review of the solutions, it is recommended
that Mathletes be provided with opportunities to present solutions to problems as preparation for the
Masters Round.
All of the problems provide students with practice in a variety of problem-solving situations and may be
used to diagnose skill levels, to practice and apply skills, or to evaluate growth in skills.
STrETCH
Page 32 contains the Transformations & Coordinate Geometry Stretch. The problems cover a variety of
difî‚¿culty levels. This Stretch, and the two included in Volume II, may be incorporated at any time.
MATHCOUNTS 2008–2009 11
ANSwErS
Answers to all problems can be found on pages 49-51.
SOLUTIONS
Complete solutions for the problems start on page 53. These are only possible solutions. It is very
possible you and/or your students will come up with more elegant solutions.
SCHEDULE
The Stretches can be used at any time. The following chart is the recommended schedule for using the
Warm-Ups and Workouts (Volumes I and II of the handbook are required to complete this schedule.):
September Warm-Ups 1–2 Workout 1
October Warm-Ups 3–6 Workouts 2–3
November Warm-Ups 7–10 Workouts 4–5
December Warm-Ups 11–14 Workouts 6– 7
January Warm-Ups 15–16 Workout 8
MATHCOUNTS School Competition
Warm-Ups 17–18 Workout 9
February Selection of competitors for Chapter Competition
MATHCOUNTS Chapter Competition
To encourage participation by the greatest number of students, postpone selection of your school’s ofcial
competitors until just before the local competition.
On average, MATHCOUNTS coaches meet with Mathletes for an hour one or two times a week at the
beginning of the year and with increasing frequency as the competitions approach. Sessions may be
held before school, during lunch, after school or on weekends—whatever works best with your school’s
schedule and limits scheduling conicts with other activities.
Some suggestions for getting the most out of the Warm-Ups and Workouts at coaching sessions:
• Encourage discussion of the problems so that students learn from one another.
• Encourage a variety of methods for solving problems.
• Have students write problems for each other.
• Use the MATHCOUNTS Problem of the Week. Based on current events, this problem is posted every
Monday on the MATHCOUNTS web site at www.mathcounts.org.
• Practice working in groups to develop teamwork (and to prepare for the Team Round).
• Practice oral presentations to reinforce understanding (and to prepare for the Masters Round).
• Take advantage of additional MATHCOUNTS coaching materials, such as previous years’
competitions, to provide an extra challenge or to prepare for competition.
• Provide refreshments and vary the location of your meetings to create a relaxing, fun atmosphere.
• Invite the school principal to a session to offer words of support.
• Volunteer assistance can be used to enrich the program and expand it to more students. Fellow
teachers can serve as assistant coaches. Individuals such as MATHCOUNTS alumni and high school
students, parents, community professionals and retirees also can help.
12 MATHCOUNTS 2008–2009
OFFICIAL rULES & PrOCEDUrES
The following rules and procedures govern all MATHCOUNTS competitions. The MATHCOUNTS
Foundation reserves the right to alter these rules and procedures at any time. Coaches are responsible
for being familiar with the rules and procedures outlined in this handbook. Coaches should bring any
difî‚¿culty in procedures or in student conduct to the immediate attention of the appropriate chapter, state
or national ofî‚¿cial. Students violating any rules may be subject to immediate disqualiî‚¿cation.
rEgISTrATION
To participate in the MATHCOUNTS Competition Program, a school representative is required to
complete and return the Request/Registration Form (available at the back of this handbook and on
our web site at www.mathcounts.org) along with a check, money order, purchase order or credit
card authorization to be postmarked no later than Dec. 12, 2008, to: MATHCOUNTS Registration,
P.O. Box 441, Annapolis Junction, MD 20701. The team registration fee is $80. The individual
registration fee is $20 per student. Reduced fees of $40 per team and $10 per individual are available to
schools entitled to receive Title I funds. Registration fees are nonrefundable.
VERY IMPORTANT NOTE: Do not hold up the mailing of your Volume II because you
are waiting for a purchase order to be processed or a check to be cut by your school for
the registration fee. Fill out your Request/Registration Form and send in a photocopy of it
without payment. We immediately will mail your Club in a Box resource kit (which contains
Volume II of the MATHCOUNTS School Handbook) and credit your account once your
payment is received.
By completing the Request/Registration Form, the coach attests to the school administration’s permission
to register students for MATHCOUNTS.
Academic centers or enrichment programs that do not function as students’ ofcial school of record are
not eligible to register.
Registration in the Competition Program entitles a school to send students to the local competition and
earns the school Bronze Level Status in the MATHCOUNTS Club Program. Additionally, registered
schools will receive two mailings.
â–ª The î‚¿rst, immediate mailing will be the Club in a Box resource kit that contains Volume II of the
2008–2009 MATHCOUNTS School Handbook, the Club Resource Guide with 12 club meeting ideas
and other materials for the MATHCOUNTS Club Program.
â–ª The second mailing will include the School Competition Kit (with instructions, School Competition
& Answer Key, recognition ribbons and student participation certiî‚¿cates), a catalog of additional
coaching materials and MATHCOUNTS News
. The î‚¿rst batch of School Competition Kits will be
mailed in early November, and additional mailings will occur on a rolling basis to schools sending in
the Request/Registration Forms later in the fall.
Your registration form must be postmarked by Dec. 12, 2008. In some circumstances, late registrations
may be accepted at the discretion of MATHCOUNTS and the local coordinator. The sooner you register,
the sooner you will receive your School Competition materials and can start preparing your team.
Once processed, conî‚¿rmation of your registration will be available through the registration database in
the Registered Schools section of the MATHCOUNTS web site (www.mathcounts.org). Other questions
about the status of your registration should be directed to: MATHCOUNTS Registration, P.O. Box 441,
Annapolis Junction, MD 20701. Telephone: 301-498-6141. Your state or local coordinator will be notiî‚¿ed
of your registration, and you then will be informed of the date and location of your local competition.
If you have not been contacted by mid-January with competition details, it is your responsibility to
contact your local coordinator to conî‚¿rm that your registration has been properly routed and that your
school’s participation is expected. Coordinator contact information is available in the Find a Coordinator
section of www.mathcounts.org. Questions speciî‚¿c to a local or state program should be addressed to the
coordinator in your area.
MATHCOUNTS 2008–2009 13
ELIgIBLE PArTICIPANTS
Students enrolled in the 6th, 7th or 8th grade are eligible to participate in MATHCOUNTS
competitions.
Students taking middle school mathematics classes who are not full-time 6th-, 7th- or
8th-graders are not eligible. Participation in MATHCOUNTS competitions is limited to three years for
each student though there is no limit to the number of years a student may participate in the school-based
coaching phase.
School Registration: A school may register one team of four and up to four individuals for a total
of eight participants. You must designate team members versus individuals prior to the start of the
local (chapter) competition (i.e., a student registered as an “individual” may not help his/her school team
advance to the next level of competition).
Team Registration: Only one team (of up to four students) per school is eligible to compete.
Members of a school team will participate in the Sprint, Target and Team Rounds. Members of a school
team also will be eligible to qualify for the Countdown Round (where conducted). Team members will
be eligible for team awards, individual awards and progression to the state and national levels based on
their individual and/or team performance. It is recommended that your strongest four Mathletes form
your school team. Teams of fewer than four will be allowed to compete; however, the team score will
be computed by dividing the sum of the team members’ scores by four (see “Scoring” on page 17 for
details
). Consequently, teams of fewer than four students will be at a disadvantage.
Individual Registration: Up to four students may be registered in addition to or in lieu of a school
team. Students registered as individuals will participate in the Sprint and Target Rounds but not the Team
Round. Individuals will be eligible to qualify for the Countdown Round (where conducted). Individuals
also will be eligible for individual awards and progression to the state and national levels.
School Denitions: Academic centers or enrichment programs that do not function as students’ ofcial
school of record are not eligible to register. If it is unclear whether an educational institution is considered
a school, please contact your local Department of Education for speciî‚¿c criteria governing your state.
School Enrollment Status: A student may compete only for his/her ofcial school of record. A student’s
school of record is the student’s base or main school. A student taking limited course work at a second
school or educational center may not register or compete for that second school or center, even if the
student is not competing for his/her school of record. MATHCOUNTS registration is not determined by
where a student takes his/her math course. If there is any doubt about a student’s school of record, the
local or state coordinator must be contacted for a decision before registering.
Small Schools:
Schools with eight or fewer students in each of the 6th, 7th and 8th grades are permitted
to combine to form a MATHCOUNTS team. Only schools from the same or adjacent chapters within a
state may combine to form a team. The combined team will compete in the chapter where the coach’s
school is located.
Homeschools: Homeschools in compliance with the homeschool laws of the state in which they are
located are eligible to participate in MATHCOUNTS competitions in accordance with all other rules.
Homeschool coaches must complete an afî‚¿davit verifying that students from the homeschool are in the
6th, 7th or 8th grade and that the homeschool complies with applicable state laws. Completed afî‚¿davits
must be submitted to the local coordinator prior to competition.
Virtual Schools: Any virtual school interested in registering students must contact the MATHCOUNTS
national ofî‚¿ce at 703-299-9006 before Dec. 12, 2008, for registration details.
Substitutions by Coaches: Coaches may not substitute team members for the State Competition unless
a student voluntarily releases his/her position on the school team. Additional restrictions on substitutions
(such as requiring parental release or requiring the substitution request to be submitted in writing) are at
the discretion of the state coordinator. Coaches may not make substitutions for students progressing to the
state competition as individuals. At all levels of competition, student substitutions are not permitted after
on-site competition registration has been completed. The student being added to the team need not be a
14 MATHCOUNTS 2008–2009
student who was registered for the Chapter Competition as an individual.
Religious Observances: A student who is unable to attend a competition due to religious observances
may take the written portion of the competition up to one week in advance of the scheduled competition.
In addition, all competitors from that school must take the exam at the same time. Advance testing will
be done at the discretion of the local and state coordinators and under proctored conditions. If the student
who is unable to attend the competition due to a religious observance is not part of the school team,
then the team has the option of taking the Team Round during this advance testing or on the regularly
scheduled day of the competition with the other teams. The coordinator must be made aware of the team’s
decision before the advance testing takes place. Students who qualify for an ofî‚¿cial Countdown Round
but are unable to attend will automatically forfeit one place standing.
Special Needs: Reasonable accommodations may be made to allow students with special needs to
participate. Requests for accommodation of special needs must be directed to local or state coordinators
in writing at least three weeks in advance of the local or state competition. This written request should
thoroughly explain the student’s special need as well as what the desired accommodation would entail.
Many accommodations that are employed in a classroom or teaching environment cannot be implemented
in the competition setting. Accommodations that are not permissible include, but are not limited to, granting
a student extra time during any of the competition rounds or allowing a student to use a calculator for
the Sprint or Countdown Rounds. In conjunction with the MATHCOUNTS Foundation, coordinators
will review the needs of the student and determine if any accommodations will be made. In making î‚¿nal
determinations, the feasibility of accommodating these needs at the National Competition will be taken into
consideration.
LEvELS OF COMPETITION
MATHCOUNTS competitions are organized at four levels: school, chapter (local), state and national.
Competitions are written for the 6th- through 8th-grade audience. The competitions can be quite
challenging, particularly for students who have not been coached using MATHCOUNTS materials. All
competition materials are prepared by the national ofî‚¿ce.
The real success of MATHCOUNTS is inuenced by the coaching sessions at the school level. This
component of the program involves the most students (more than 500,000 annually), comprises the
longest period of time and demands the greatest involvement.
SCHOOL COMPETITION: In January, after several months of coaching, schools registered for the
Competition Program should administer the School Competition to all interested students. The School
Competition is intended to be an aid to the coach in determining competitors for the Chapter (local)
Competition. Selection of team and individual competitors is entirely at the discretion of coaches and
need not be based solely on School Competition scores. The School Competition is sent to the coach
of a school, and it may be used by the teachers and students only in association with that school’s
programs and activities. The current year’s School Competition questions must remain condential
and may not be used in outside activities, such as tutoring sessions or enrichment programs with
students from other schools. For additional announcements or edits, please check the “Coaches’
Forum” on the MATHCOUNTS web site before administering the School Competition.
It is important that the coach look upon coaching sessions during the academic year as opportunities to
develop better math skills in all students, not just in those students who will be competing. Therefore, it is
suggested that the coach postpone selection of competitors until just prior to the local competitions.
CHAPTER COMPETITIONS: Held from Feb. 1 through Feb. 28, 2009, the Chapter Competition
consists of the Sprint, Target and Team Rounds. The Countdown Round (ofî‚¿cial or just for fun) may or
may not be included. The chapter and state coordinators determine the date and administration of the local
competition in accordance with established national procedures and rules. Winning teams and students
will receive recognition. The winning team will advance to the State Competition. Additionally, the two
highest-ranking competitors not on the winning team (who may be registered as individuals or as members
of a team) will advance to the State Competition. This is a minimum of six advancing Mathletes (assuming
MATHCOUNTS 2008–2009 1
the winning team has four members). Additional teams and/or Mathletes also may progress at the discretion
of the state coordinator. The policy for progression must be consistent for all chapters within a state.
STATE COMPETITIONS: Held from March 1 through March 28, 2009, the State Competition
consists of the Sprint, Target and Team Rounds. The Countdown Round (ofî‚¿cial or just for fun) and
the optional Masters Round may or may not be included. The state coordinator determines the date and
administration of the State Competition in accordance with established national procedures and rules.
Winning teams and students will receive recognition. The four highest-ranked Mathletes and the coach of
the winning team from each State Competition will receive an all-expenses-paid trip to the National Competition.
RAYTHEON MATHCOUNTS NATIONAL COMPETITION: Held Friday, May 8, 2009, in
Orlando at the Walt Disney World Swan and Dolphin Resort, the National Competition consists
of the Sprint, Target, Team, Countdown and Masters Rounds. Expenses of the state team and coach to
travel to the National Competition will be paid by MATHCOUNTS. The national program does not make
provisions for the attendance of additional students or coaches. All national competitors will receive a
plaque and other items in recognition of their achievements. Winning teams and individuals also will
receive medals, trophies and college scholarships.
COMPETITION COMPONENTS
MATHCOUNTS competitions are designed to be completed in approximately three hours:
The
SPRINT ROUND (40 minutes) consists of 30 problems. This round tests accuracy, with time being
such that only the most capable students will complete all of the problems. Calculators are not permitted.
The
TARGET ROUND (approximately 30 minutes) consists of eight problems presented to competitors
in four pairs (6 minutes per pair). This round features multi-step problems that engage Mathletes in
mathematical reasoning and problem-solving processes. Problems assume the use of calculators.
The
TEAM ROUND (20 minutes) consists of 10 problems that team members work together to solve.
Team member interaction is permitted and encouraged. Problems assume the use of calculators.
Note: Coordinators may opt to allow those competing as “individuals” to create a “squad” of four to take
the Team Round for the experience, but the round should not be scored and is not considered ofî‚¿cial.
The
COUNTDOWN ROUND is a fast-paced, oral competition for top-scoring individuals (based on
scores in the Sprint and Target Rounds). In this round, pairs of Mathletes compete against each other and
the clock to solve problems. Calculators are not permitted.
At Chapter and State Competitions, a Countdown Round may be conducted ofî‚¿cially, unofî‚¿cially (for
fun) or omitted. However, the use of an ofî‚¿cial Countdown Round will be consistent for all chapters
within a state. In other words, all chapters within a state must use the round ofî‚¿cially in order for any
chapter within a state to use it ofî‚¿cially. All students, whether registered as part of a school team or as an
individual competitor, are eligible to qualify for the Countdown Round.
An ofcial Countdown Round is dened as one that determines an individual’s nal overall rank in
the competition. If the Countdown Round is used ofî‚¿cially, the ofî‚¿cial procedures as established by the
MATHCOUNTS Foundation must be followed.
If a Countdown Round is conducted unofî‚¿cially, the ofî‚¿cial procedures do not have to be followed.
Chapters and states choosing not to conduct the round ofî‚¿cially must determine individual winners on the
sole basis of students’ scores in the Sprint and Target Rounds of the competition.
In an ofî‚¿cial Countdown Round, the top 25% of students, up to a maximum of 10, are selected to
compete. These students are chosen based on their individual scores. The two lowest-ranked students are
paired, a question is projected and students are given 45 seconds to solve the problem. A student may buzz
in at any time, and if s/he answers correctly, a point is scored; if a student answers incorrectly, the other
student has the remainder of the 45 seconds to answer. Three questions are read to each pair of students,
one question at a time, and the student who scores the most points (not necessarily 2 out of 3) captures
the place, progresses to the next round and challenges the next highest-ranked student. (If students are
16 MATHCOUNTS 2008–2009
tied after three questions [at 1-1 or 0-0], questions continue to be read until one is successfully answered.)
This procedure continues until the fourth-ranked Mathlete and her/his opponent compete. For the î‚¿nal
four rounds, the î‚¿rst student to correctly answer three questions advances. The Countdown Round
proceeds until a î‚¿rst-place individual is identiî‚¿ed. (More detailed rules regarding the Countdown Round
procedure are identied in the “Instructions” section of the School Competition Booklet.)
Note: Rules for the Countdown Round change for the National Competition.
The Masters Round is a special round for top individual scorers at the state and national levels. In this
round, top individual scorers prepare an oral presentation on a speciî‚¿c topic to be presented to a panel
of judges. The Masters Round is optional at the state level; if held, the state coordinator determines the
number of Mathletes that participate. At the national level, four Mathletes participate. (Participation in the
Masters Round is optional. A student declining to compete will not be penalized.)
Each student is given 30 minutes to prepare his/her presentation. Calculators may be used. The
presentation will be 15 minutes—up to 11 minutes may be used for the student’s oral response to the
problem, and the remaining time may be used for questions by the judges. This competition values
creativity and oral expression as well as mathematical accuracy. Judging of presentations is based on
knowledge, presentation and the responses to judges’ questions.
ADDITIONAL rULES
All answers must be legible.
Pencils and paper will be provided for Mathletes by competition organizers. However, students may
bring their own pencils, pens and erasers if they wish. They may not use their own scratch paper.
Use of notes or other reference materials (including dictionaries) is not permitted.
Speciî‚¿c instructions stated in a given problem take precedence over any general rule or procedure.
Communication with coaches is prohibited during rounds but is permitted during breaks. All
communication between guests and Mathletes is prohibited during competition rounds. Communication
between teammates is permitted only during the Team Round.
Calculators are not permitted in the Sprint or Countdown Rounds, but they are permitted in
the Target, Team and Masters Rounds. Where calculators are permitted, students may use any
calculator (including programmable and graphing calculators) that does not contain a QWERTY (i.e.,
typewriter-like) keypad. Calculators that have the ability to enter letters of the alphabet but do not have
a keypad in a standard typewriter arrangement are acceptable. Personal digital assistants (e.g., Palm
Pilots
®
) are not considered to be calculators and may not be used during competitions. Students
may not use calculators to exchange information with another person or device during the
competition.
Coaches are responsible for ensuring that their students use acceptable calculators, and students
are responsible for providing their own calculators. Coordinators are not responsible for providing
Mathletes with calculators, AC outlets or batteries before or during MATHCOUNTS competitions.
Coaches are strongly advised to bring backup calculators and spare batteries to the competition for their
team members in case of a malfunctioning calculator or weak/dead batteries. Neither the MATHCOUNTS
Foundation nor coordinators shall be responsible for the consequences of a calculator’s malfunctioning.
Pagers, cell phones, radios and MP3 players should not be brought into the competition room.
Failure to comply could result in dismissal from the competition.
Should there be a rule violation or suspicion of irregularities, the MATHCOUNTS coordinator or
competition ofî‚¿cial has the obligation and authority to exercise his/her judgment regarding the situation
and take appropriate action, which might include disqualiî‚¿cation of the suspected student(s) from the
competition.
MATHCOUNTS 2008–2009 17
SCOrINg
Scores on the competition do not conform to traditional grading scales. Coaches and students should
view an individual written competition score of 23 (out of a possible 46) as highly commendable.
The individual score is the sum of the number of Sprint Round questions answered correctly and twice
the number of Target Round questions answered correctly. There are 30 questions in the Sprint Round and
8 questions in the Target Round, so the maximum possible individual score is 30 + 2(8) = 46.
The team score
is calculated by dividing the sum of the team members’ individual scores by 4 (even if
the team has fewer than four members) and adding twice the number of Team Round questions answered
correctly. The highest possible individual score is 46. Four students may compete on a team, and there
are 10 questions in the Team Round. Therefore, the maximum possible team score is
((46 + 46 + 46 + 46) ÷ 4) + 2(10) = 66.
If used ofî‚¿cially, the Countdown Round yields î‚¿nal individual standings. The Masters Round is a
competition for the top-scoring individuals that yields a separate winner and has no impact on progression
to the National Competition.
Ties will be broken as necessary to determine team and individual prizes and to determine which
individuals qualify for the Countdown Round. For ties among individuals, the student with the higher
Sprint Round score will receive the higher rank. If a tie remains after this comparison, speciî‚¿c groups
of questions from the Sprint and Target Rounds are compared. For ties among teams, the team with the
higher Team Round score, and then the higher sum of the team members’ Sprint Round scores, receives
the higher rank. If a tie remains after these comparisons, speciî‚¿c questions from the T
eam Round
will be compared. Note: These are very general guidelines. Please refer to the “General Instructions”
accompanying each competition set for detailed procedures should a tie occur.
In general, questions in the Sprint, Target and Team Rounds increase in difî‚¿culty so that the most difî‚¿cult
questions occur near the end of each round. The comparison of questions to break ties generally occurs
such that those who correctly answer the more difî‚¿cult questions receive the higher rank.
Protests concerning the correctness of an answer on the written portion of the competition must
be registered with the room supervisor in writing by a coach within 30 minutes of the end of each
round. Rulings on protests are î‚¿nal and may not be appealed. Protests will not be accepted during the
Countdown or Masters Rounds.
rESULTS DISTrIBUTION
Coaches should expect to receive the scores of their students and a list of the top 25% of students and
top 40% of teams from their coordinator. In addition, single copies of the blank competition materials
and answer keys may be distributed to coaches after all competitions at that level nationwide have been
completed. Coordinators must wait for veriî‚¿cation from the national ofî‚¿ce that all such competitions
have been completed before distributing blank competition materials and answer keys. Both the
problems and answers from Chapter and State competitions will be posted on the MATHCOUNTS web
site following the completion of all competitions at that level nationwide (Chapter – early March;
State – early April). The previous year’s problems and answers will be taken off the web site at that time.
Student competition papers and answers will not be viewed by nor distributed to coaches, parents,
students or other individuals. Students’ competition papers become the condential property of the
MATHCOUNTS Foundation.
18 MATHCOUNTS 2008–2009
FOrMS OF ANSwErS
The following list explains acceptable forms for answers. Coaches should ensure that Mathletes are
familiar with these rules prior to participating at any level of competition. Judges will score competition
answers in compliance with these rules for forms of answers.
All answers must be expressed in simplest form. A “common fraction” is to be considered a fraction
in the form ±
a
b
, where a and b are natural numbers and GCF(a, b) = 1. In some cases the term “common
fraction” is to be considered a fraction in the form
A
B
, where A and B are algebraic expressions and A and
B do not share a common factor. A simplied “mixed number” (“mixed numeral,” “mixed fraction”) is to
be considered a fraction in the form ± N
a
b
, where N, a and b are natural numbers, a < b and GCF(a, b) = 1.
Examples:
Problem:
Express 8 divided by 12 as a common fraction. Answer:
2
3
Unacceptable:
4
6
Problem: Express 12 divided by 8 as a common fraction. Answer:
3
2
Unacceptable:
12 1
8 2
, 1
Problem: Express the sum of the lengths of the radius and the circumference of a circle with a diameter
of
1
4
as a common fraction in terms of π. Answer:
1 2
8
+ π
Problem: Express 20 divided by 12 as a mixed number. Answer:
2
3
Unacceptable:
8 5
12 3
,
Ratios should be expressed as simpliî‚¿ed common fractions unless otherwise speciî‚¿ed. Examples:
Simpliî‚¿ed, Acceptable Forms:
7 3 4
2 6
, ,
−π
Ï€
Unacceptable:
1
4
1
2 3
,
, 3.5, 2:1
Radicals must be simpliî‚¿ed. A simpliî‚¿ed radical must satisfy: 1) no radicands have a factor which
possesses the root indicated by the index; 2) no radicands contain fractions; and 3) no radicals appear in
the denominator of a fraction. Numbers with fractional exponents are not in radical form. Examples:
Problem: Evaluate
15 5×
. Answer:
5 3
Unacceptable:
75
Answers to problems asking for a response in the form of a dollar amount or an unspeciî‚¿ed
monetary unit (e.g., “How many dollars...,” “How much will it cost...,” “What is the amount of
interest...”) should be expressed in the form ($) a.bc, where a is an integer and b and c are digits.
The only exceptions to this rule are when a is zero, in which case it may be omitted, or when b and c are
both zero, in which case they may both be omitted. Examples:
Acceptable:
2.35, 0.38, .38, 5.00, 5 Unacceptable: 4.9, 8.0
Units of measurement are not required in answers, but they must be correct if given. When a
problem asks for an answer expressed in a speciî‚¿c unit of measure or when a unit of measure is provided
in the answer blank, equivalent answers expressed in other units are not acceptable. For example, if a
problem asks for the number of ounces and 36 oz is the correct answer, 2 lbs 4 oz will not be accepted. If
a problem asks for the number of cents and 25 cents is the correct answer, $0.25 will not be accepted.
Do not make approximations for numbers (e.g., π,
2
3
,
5 3
) in the data given or in solutions unless
the problem says to do so.
Do not do any intermediate rounding (other than the “rounding” a calculator performs) when
calculating solutions. All rounding should be done at the end of the calculation process.
Scientic notation should be expressed in the form a × 10
n
where a is a decimal, 1 < |a| < 10, and n is an
integer. Examples:
Problem:
Write 6895 in scientic notation. Answer: 6.895 × 10
3
Problem: Write 40,000 in scientic notation. Answer: 4 × 10
4
or 4.0 × 10
4
An answer expressed to a greater or lesser degree of accuracy than called for in the problem will not
be accepted. Whole number answers should be expressed in their whole number form.
Thus, 25.0 will not be accepted for 25 nor vice versa.
The plural form of the units will always be provided in the answer blank, even if the answer
appears to require the singular form of the units.
1
1
3
MATHCOUNTS 2008–2009 19
MATHCOUNTS recognizes that math clubs can play an important role
in shaping students’ attitudes and abilities. In an effort to support existing
math clubs and their coaches, as well as encourage the formation of new math
clubs, MATHCOUNTS offers the MATHCOUNTS Club Program (MCP).
Whether you are starting a new program or continuing a tradition of strong math clubs in your school,
MATHCOUNTS understands the challenges involved in such a commitment. MATHCOUNTS also
understands the meaningful rewards of coaching a math club and the strong impact you can have on
students through these organized activities. The MATHCOUNTS Club Program is designed to provide
schools with the structure and activities to hold regular meetings of a math club. Depending on the level
of student and teacher involvement, a school may receive a recognition plaque or banner and be entered
into a drawing for prizes.
This program may be used by schools as a stand-alone program or incorporated into the student
preparation for the MATHCOUNTS Competition Program.
CLUB MATErIALS
When a school signs up a club in the MATHCOUNTS Club Program, the school will be sent a Club in a
Box resource kit containing (1) further details on the Club Program, (2) the Club Resource Guide, which
outlines structured club activities, (3) the î‚¿rst monthly math challenge, (4) BINGO cards to accompany
one of the December activities in the Club Resource Guide, (5) 12 MATHCOUNTS pencils and (6) a
MATHCOUNTS car magnet for the coach. Additionally, î‚¿ve other monthly math challenges and an
Ultimate Math Challenge will be made available online for use by math club students.
The Club Resource Guide and the monthly math challenges are the backbone of the MCP. The
Club
Resource Guide
contains 12 meeting/activity ideas you can use as a basis for planning fun, instructive
get-togethers for your math club throughout the year. Each of the meeting themes was written with a
particular time of the year or holiday in mind, but there is no particular order in which the meeting ideas
must be done, and meeting plans certainly do not have to be used on the exact holiday referenced. In
addition to the meeting ideas, six monthly math challenges are provided. Club sponsors should work these
monthly math challenges into their meeting plans, as completion of these challenges is required for Silver
Level Status.
In addition to the materials sent in the Club in a Box resource kit, a special MCP Members Only page in
the Club Program section of www.mathcounts.org will become visible and available to any coach signing
up a club in the MCP or registering students in the Competition Program. (Coaches also must create a
User Proî‚¿le on mathcounts.org to gain access to this members-only page.) This web page provides î‚¿ve
more monthly math challenges (released one per month), 13 meeting plan ideas from last year and all of
the resources necessary to conduct any of the suggested meeting ideas.
rULES, PrOCEDUrES & DEADLINES
gETTINg STArTED
The MATHCOUNTS Club Program is open only to 6th-, 7th- and 8th-grade students in U.S.- based
schools.
The club coach must complete and submit the MATHCOUNTS Request/Registration Form to sign
up the school’s math club. By selecting Option 1 or Option 2 on this form, your school has reached
1.
2.
MATHCOUNTS CLUB
PrOgrAM...
A MOrE DETAILED LOOk
20 MATHCOUNTS 2008–2009
Bronze Level Status and will be recognized on the MATHCOUNTS web site. (The MATHCOUNTS
Request/Registration Form is available at the back of this book and online at www.mathcounts.org.)
Shortly after MATHCOUNTS receives your Request/Registration Form, we will send you Volume II
of the 2008–2009 MATHCOUNTS School Handbook and the Club in a Box resource kit.
Begin recruiting club members and spreading the word about your î‚¿rst club meeting. The Club
Resource Guide included in the Club in a Box resource kit contains many helpful ideas for starting a
club program.
Start using the handbook problems and materials in the Club in a Box resource kit with the students
in your math club. Among other items, the resource kit includes structured club activities, the î‚¿rst of
six monthly math challenges, pencils for your students and a set of BINGO cards that accompanies
one of the December activities in the Club Resource Guide.
ATTAININg SILvEr LEvEL STATUS
Though we hope more than 12 students will participate in your math club and tackle the monthly
math challenges, your school must have at least 12 students who each participate in at least 5 of
the 6 monthly math challenges. A new monthly math challenge will be available each month from
September through February.
Each of the monthly math challenges (and their answer keys) will be available on the MCP Members
Only page in the Club Program section of www.mathcounts.org.
Once your school club has 12 students who have each fulî‚¿lled the Silver Level requirement of
completing at least 5 of the 6 monthly math challenges, complete the Participant V
eriî‚¿cation Form/
Application for Silver Level Status with the names of your 12 students and your contact information.
This form is available in the Club in a Box resource kit, the Club Resource Guide (page 15) and the
MCP Members Only page of the MATHCOUNTS web site.
Submit your Participant Veriî‚¿cation Form/Application for Silver
Level Status via fax or mail to the address shown here. (Please
submit your form only once.)
Deadline: Your Participant Veriî‚¿cation Form/Application for Silver Level Status must be
received by March 6, 2009 for your school to be eligible for the prize drawing ($250 gift cards).
The winners of the drawing will be notiî‚¿ed by April 17, 2009. A list of the winners also will be
posted online. MATHCOUNTS will send the prizes to the winners by April 30, 2009.
In May, MATHCOUNTS will send a plaque to all Silver Level Schools in recognition of their
achievement.
ATTAININg gOLD LEvEL STATUS
Your math club î‚¿rst must attain Silver Level Status in the Club Program.
Once reaching Silver Level Status, MATHCOUNTS will e-mail the coach the Ultimate Math
Challenge. (The î‚¿rst e-mails will go out Feb. 13, 2009. Schools attaining Silver Level Status after
this date will receive their Ultimate Math Challenge within one week of attaining Silver Level
Status.)
3.
4.
5.
1.
2.
3.
4.
5.
6.
7.
1.
2.
MATHCOUNTS Foundation
Silver Level – Club Program
1420 King Street
Alexandria, VA 22314
Fax: 703-299-5009
MATHCOUNTS 2008–2009 21
MATHCOUNTS Foundation
Gold Level – Club Program
1420 King Street
Alexandria, VA 22314
Once your students have completed the Ultimate Math Challenge,
mail students’ completed challenges (max. of 20 per school) and
the Veriî‚¿cation Form/Application for Gold Level Status to the
address shown here. More speciî‚¿c details and the Veriî‚¿cation
Form/Application for Gold Level Status will be provided when the
Ultimate Math Challenge is e-mailed to the coach.
MATHCOUNTS will score the students’ papers and determine if your math club has attained Gold
Level Status. Every math club member may take the Ultimate Math Challenge. However, no more
than 20 completed challenges per school may be submitted. At least 12 of the submitted challenges
each must have 80% or more of the problems answered correctly for a school to attain Gold Level
Status.
Deadline: Your Verication Form/Application for Gold Level Status and your students’
completed challenges must be received by March 27, 2009 for your school to be eligible for the
prize drawing ($500 gift cards/trip to the National Competition).
The winners of the drawing will be notiî‚¿ed by April 30,
2009. A list of the winners also will be posted online.
MATHCOUNTS will send the prizes to the winners by April
30, 2009.
In May, MATHCOUNTS will send a banner and a plaque to
all Gold Level Schools in recognition of their achievement.
(Gold Level Schools will not receive the Silver Level plaque,
too, but will be entered into the Silver Level drawing for
prizes.)
FrEqUENTLY ASkED qUESTIONS
Who is eligible to participate?
Anyone eligible for the MATHCOUNTS Competition Program is eligible to participate in the Club
Program. The Club Program is open to all U.S. schools with 6th-, 7th- and/or 8th-grade students.
Schools with 12 or fewer students in each of the 6th, 7th and 8th grades are permitted to combine for the
purpose of reaching Silver or Gold Level Status. Similarly, homeschools may combine for the purpose of
reaching Silver or Gold Level Status. See page 13 of the MATHCOUNTS School Handbook for details on
eligibility for the MATHCOUNTS Competition Program.
How many students can participate?
There is no limit to the number of students who may participate in the MATHCOUNTS Club Program.
Encourage every interested 6th-, 7th- and/or 8th-grade student to get involved.
What if our school has more than one math club?
MATHCOUNTS encourages all math clubs in a school to make use of the MATHCOUNTS Club Program
materials. However, each school may have only one ofî‚¿cially registered club in the MATHCOUNTS
Club Program. Therefore, it is recommended that schools combine their clubs when working toward
meeting the requirements of Silver or Gold Level Status.
What does it cost to participate?
Nothing. There is no fee to participate in the Club Program. Similar to the MATHCOUNTS School
Handbook, the Club in a Box and other resources are free for all eligible schools that request them.
Can a school participate in both the Club Program and the Competition Program?
YES. A school may choose to participate in the Club Program, the Competition Program or both.
3.
4.
5.
6.
7.
The Gold Level grand prize drawing winners
received their Gold Level banner at the
Lockheed Martin MATHCOUNTS National
Competition - 2008.
22 MATHCOUNTS 2008–2009
Since these programs can complement each other, any school that registers for the MATHCOUNTS
Competition Program (Option 2 on the MATHCOUNTS Request/Registration Form) will automatically
be signed up for the Club Program and sent the Club in a Box resource kit.
How is the Club Program different from the Competition Program?
The Club Program does not include a school versus school competition with the opportunity for top
performers to advance. There are no fees to participate in the Club Program, and recognition is focused
entirely on the school and math club.
What are the different levels and requirements of the program?
Level Requirements School Receives
BRONZE
1. On the MATHCOUNTS Request/
Registration Form (page 63) or online at
www.mathcounts.org, choose
Option 1 (Club Program) or
Option 2 (Club Program and
Competition Program) and î‚¿ll in the
required information.
2. Submit the form to MATHCOUNTS.
♦ Recognition on www.mathcounts.org
♦ Club in a Box resource kit
♦ Volume II of the 2008–2009
MATHCOUNTS School Handbook,
containing 200 math problems
SILVER
1. At least 12 members of the math club
each must take at least 5 of the 6 monthly
math challenges (available online
September through February).
2. The Participant Veriî‚¿cation Form/
Application for Silver Level Status must
be received by MATHCOUNTS by
March 6, 2009 (available in the Club in
a Box resource kit, the Club Resource
Guide and the MCP Members Only
page in the Club Program section of
www.mathcounts.org).
♦ Recognition on www.mathcounts.org
♦ Certicate for students
♦ Plaque identifying your school as a Silver
Level MATHCOUNTS School
♦ Entry into a drawing for one of ten $250
gift cards for student recognition (awards/
party)
*Schools going on to reach Gold Level Status will be
included in the Silver Level drawing, but will receive
only the Gold Level plaque.
GOLD
1. Achieve Silver Level Status.
2. At least 12 members of the math club
each must score an 80% or higher on the
Ultimate Math Challenge (available in
February/e-mailed to coaches of Silver
Level Schools).
3. The completed Ultimate Math
Challenges and Veriî‚¿cation Form/
Application for Gold Level Status
must be received by MATHCOUNTS
by March 27, 2009. (Any number of
students may take the challenge, but a
maximum of 20 completed Ultimate
Challenges may be submitted per
school.)
♦ Recognition on www.mathcounts.org
♦ Certicate for students
♦ Banner and plaque identifying your
school as a Gold Level MATHCOUNTS
School
♦ Entry into a drawing for:
1) One of î‚¿ve $500 gift cards for student
recognition (awards/party)
2) Grand Prize: $500 gift card for student
recognition (awards/party) and a trip
for four students and the coach to
witness the Raytheon MATHCOUNTS
National Competition at the Walt
Disney World Swan and Dolphin
Resort (May 7-10, 2009)
MATHCOUNTS reserves the right to modify the MATHCOUNTS Club Program guidelines as necessary.
MATHCOUNTS 2008-2009 23
Warm-Up 1
1. _________ Yoselin purchases 3 dozen tomatoes for $6.66. At this rate, how
much will 10 dozen tomatoes cost?
2. _________ One square has a perimeter of 40 inches. A second square has a perimeter of
36 inches. What is the positive difference in the areas of the two squares?
3. _________ A standard six-sided die was rolled 50 times, and
the outcomes are in the table shown. What percent
of the rolls resulted in a prime number?
4. _________ How many factors of 1000 can be divided by 20 without a remainder?
5. _________ The square shown is divided into 4 congruent rectangles. If the
perimeter of the square is 144 units, what is the perimeter of one
of the four congruent rectangles?
6. _________  Twointegershaveadifferenceof−18andasumof2.Whatistheproductofthe
two integers?
7. _________  Themedianofasetofconsecutiveoddintegersis138.Ifthegreatestintegerin
the set is 145, what is the smallest integer in the set?
8. _________ Among all three-digit integers from 100 to 400, how many have exactly one digit
thatisan8?
9. _________ BenandDanaretwoofthemembersontheschool’schessteam.Ina
tournament against their rival team, Ben played exactly 1 out of every 4
games. Dan, who played more games, played 14 games. What is the largest
number of games the team could have played?
10. ________  PeterPedalsrodehisbikeatotalof500milesinvedays.
Each day he rode 10 more miles than he had ridden on the
previousday.HowmanymilesdidPeterrideonjustthefth
day?
sq in
Outcome # of Occurrences
1 14
2 5
3 9
4 7
5 7
6 8
%
units
miles
$
factors
integers
games
MATHCOUNTS 2008-200924
Warm-Up 2
1. _________ A pattern of equilateral triangles will be made from
matchsticks, as shown. One whole matchstick is used
per side on each triangle. If the pattern is extended
and uses exactly 77 matchsticks, how many triangles
will be formed?
2. _________ The ratio of the length of a rectangular room to its width is 5:3. The perimeter of
theroomis48feet.Whatistheareaoftheroom?
3. _________ What is the greatest perfect square that is a factor of 7! ?
4. _________ A standard six-sided die with its faces numbered 1 to 6 is rolled once, and a dime
is tossed once. What is the probability of rolling a number less than 3 and tossing a
tail? Express your answer as a common fraction.
5. _________ In a recent survey of 300 students, 152 students had at least one dog,
120 students had at least one cat, and 46 students had at least one cat
and at least one dog. How many of the surveyed students did not
have either a cat or a dog?
6. _________  Alicia’saveragescoreonhervetestsis88points.Thescorerangeforeachtest
is 0 points to 100 points, inclusive. What is the lowest possible score that Alicia
couldhaveearnedononeofthevetests?    
7. _________  Thenumericalvalueofaparticularsquare’sareaisequaltothenumericalvalueof
its perimeter. What is the length of a side of the square?
8. _________ When converted to be in the same unit of measure, what is the ratio of 4 cm to
1 km? Express your answer as a common fraction.
9. _________ In 30 years, Sue will be 4 times as old as she is now. How old is she now?
10. ________  Alegal-sizedpieceofpapermeasures8.5inchesby14inches.Aone-inchborderof
paper is cut off from each of the four sides. How many square inches have been cut
off?
...
triangles
students
units
years
old
sq inches
points
sq feet
MATHCOUNTS 2008-2009 25
Workout 1
1. _________ A book is opened to a page at random. The product of the facing page numbers is
9,312. What is the sum of the facing page numbers?
2. _________ Henrywalkedonaateld9metersduenorthfromatree.He
then turned due east and walked 24 feet. He then turned due
south and walked 9 meters plus 32 feet. How many feet away
from his original starting point is Henry?
3. _________  Becauseofredistricting,LibertyMiddleSchool’senrollmentincreased
to598students.Thisisanincreaseof4%overlastyear’senrollment.
Whatwaslastyear’senrollment?
4. _________ John Lighthouse makes stained glass lamps. John spent $5000 to set
up his business. It costs John $30 to make each lamp. If John plans to
sell the lamps for $70 each, how many lamps would John have to sell to
recover the $5000 set-up cost and the cost involved in making each of the
lamps that he sold?
5. _________ A right triangle has a side length of 21 inches and a hypotenuse of 29 inches. A
secondtriangleissimilartotherstandhasahypotenuseof87inches.Whatis
the length of the shortest side of the second triangle?
6. _________ Let the function
R
¤
S
bedenedas
R
¤
S =
2
R
+
S
2
.Forinstance,2¤3wouldhave
a value of (2 × 2 + 3
2
),whichyieldsavalueof13.Whatisthevalueof4¤−1?
7. _________ If n > 1, what is the smallest positive integer n such that the expression
+ + + +1 2 3 ... n
simpliestoaninteger?
8. _________ Triangle ABC has three different integer side lengths. Side AC is the longest side,
andsideABistheshortestside.IftheperimeterofABCis384units,whatisthe
greatestpossibledifferenceAC−AB? 
9. _________ TheslopeofMary’slineis
7
8
.TheslopeofSue’slineisalsopositive,islessthan
1andissteeperthanMary’sline.IftheslopeofSue’slinecanbewrittenasa
common fraction with a one-digit numerator and a one-digit denominator, what is
theslopeofSue’sline?Expressyouranswerasacommonfraction.
10. ________ What is the value of
( )
( )
(
)
( ) ( )
1 1 1 1 1
2 3 5 5
0
4
1 1 1 1 ... 1− − − − −
? Express your answer as a
common fraction.
students
lamps
inches
units
feet
MATHCOUNTS 2008-200926
Warm-Up 3
1. _________  AnglesAandBaresupplementary.IfthemeasureofangleAis8timesangleB,
what is the measure of angle A?
2. _________  Inacertainsequenceofnumbers,eachnumberaftertherstis3lessthantwice
thepreviousnumber.Ifthethirdnumberinthesequenceis51,whatistherst
number of the sequence?
3. _________ Raymond buys items to sell in his store. He prices each item to be 25%
more than the wholesale cost. What price should he put on an item with a
wholesale cost of $39.00?
4. _________ Starting at the town of Euler and traveling 40 miles to the town of
Pythagoras, Rashid travels at the rate of 2 miles every 15 minutes.
Returning from Pythagoras to Euler, he travels 2 miles every 3
minutes.WhatwasRashid’saveragespeed,inmilesperhour,forthe
entire trip? Express your answer as a mixed number.
5. _________ The ratio of the length of the sides of square ABCD to the length of the sides
of square PQRS is 3:1. If the area of square ABCD is 9 square units, what is the
length of side PS?
6. _________ Seven cards each containing one of the following letters C, B, T, A, E, M and H are
placed in a hat. Each letter is used only once. Stu will pull four cards out at random
and without replacement. What is the probability that Stu pulls out M, A, T, H in
this order? Express your answer as a common fraction.
7. _________ If
x
and
y
are positive integers with
x
+
y
< 40, what is the largest possible product
xy
?
8. _________  Considertherectangularregionwithverticesat(5,4),(−5,4),(−5,−4)and(5,−4).
How many points with integer coordinates will be strictly in the interior of this
rectangular region?
9. _________ A ball is dropped straight down from a height of 16 feet. If it
bounces back each time to a height one-half the height from which
itlastfell,howfarwilltheballhavetraveledwhenithitstheoor
for the sixth time?
10. ________ What is the degree measure of an interior angle of a regular pentagon?
degrees
mph
points
feet
degrees
$
units
MATHCOUNTS 2008-2009 27
Warm-Up 4
1. _________ The day after the school election it was reported that Danny won the election for
class president. For every three votes his opponent received, Danny received
vevotes.Everystudentvotedexactlyonce.Ifstudentsvotedonlyforeither
DannyorhisopponentandDanny’sopponentreceived312votes,howmanystudents
voted in the election?
2. _________ Charlie has a basket full of fruit. The numbers of oranges, apples and bananas are
in the ratio of 1:2:3, respectively. If Charlie has 15 total apples and bananas, how
many oranges does he have?
3. _________  Whatisthesumoftheintegersfrom−30to50,inclusive?    
4. _________ The stem-and-leaf plot shows the scores of 20 students. What percent of the
students scored less than 75%? (The value 6 | 2 represents 62%.)
6 | 2 5
7 | 0 2 3 4 5
8|1123348
9 | 2 4 4 6 6 9
5. _________ How many different combinations of pennies, nickels, dimes and/or quarters result
in a sum of 35 cents?
6. _________ If the expression 3
1
× 9
2
× 27
3
×81
4
issimpliedtotheform3
m
, what is the value
of
m
?
7. _________ A group of naturalists catch, tag and release 121 trout into a lake. The next day
theycatch48trout,ofwhich22havebeentagged.Usingthisratio,howmany
trout would be estimated to be in the lake?
8. _________ By what number should you multiply
1
3
3
to get a product of 4? Express your answer
as a mixed number.
9. _________ The area of square ABCD is 36 square inches. The area of rectangle WXYZ is
20 square inches. If the two shapes have equal perimeters, what is the length of
the longer side of rectangle WXYZ?
10. ________ A farmer had a daughter who spoke in riddles. One day the child was asked to
count the number of goats and the number of ducks in the barnyard. She returned
and said, “Twice the number of heads is 76 less than the number of legs.” How
many goats were in the barnyard?
Problem#9isfromthe2007-2008MCPUltimateMathChallenge.
students
combos
trout
inches
goats
%
oranges
MATHCOUNTS 2008-200928
Workout 2
1. _________ The product of a set of consecutive integers is 720. If the mean is 3.5, what is the
sum of the integers?
2. _________ Mr. Adler and Mr. Bosch both work for the same company making $40,000 and
$38,000ayear,respectively.Theemployerwishestoraiseboththeirsalaries
for the next year so that they will be making the same amount of money
as each other. The employer will use $12,000 for the total of their
raises.BywhatpercentwillMr.Bosch’ssalaryberaised?
Express your answer to the nearest tenth.
3. _________ On a quiz of 10 questions, every correct answer earns 5 points, but every wrong
answer deducts 2 points. Questions left blank earn zero points. Tim got
c
questions
correct,
w
questions wrong and left
b
questions blank. He earned a score of 31 on
the quiz. What is the ordered triple (
c
,
w
,
b
)?
4. _________ Three consecutive primes are summed. When the sum is squared, the result is
72,361. What is the largest of the three primes?
5. _________  TheformulaforthevolumeofasphereisV=(4/3)π
r
3
, where
r
is the radius of the
sphere. Alex places a particular sphere into a cubic box with sides of 10 meters. If
thesphereistangenttoeachofthebox’sfaces,whatisthevolumeofthesphere?
Express your answer as a decimal to the nearest tenth.
6. _________  Annarantoherfriend’shouseatarateof8milesperhour.Onthewaybackshe
ranthesamerouteinreverse,butsheranatarateof6milesperhour.IfAnna’s
route is one-mile long each way, how many minutes longer did it take her to run
backfromherfriend’shousethanittookhertoruntoherfriend’shouse?Express
your answer as a decimal to the nearest tenth.
7. _________ Twenty-nine is the shortest leg of a right triangle whose other leg and hypotenuse
are consecutive whole numbers. What is the sum of the lengths of the other two
sides?
8. _________ Of all the four-digit positive integers containing only digits from the set {2, 4, 6,
8},whatfractionofthemhaveatleastoneoftheirdigitsrepeated?Expressyour
answer as a common fraction.
9. _________ Due to bowling a score of 204 in his last game, Remy raised his
averagefromexactly156toexactly158.Whatscoremusthebowlin
the next game to raise his overall average to exactly 159?
10. ________ The positive square root of 200 is what percent larger than the positive square
root of 121? Express your answer to the nearest whole number.
%
( , , )
cu meters
minutes
%
units
MATHCOUNTS 2008-2009 29
Warm-Up 5
1. _________ AlargemapoftheUnitedStatesusesascaleof2cm=2.5km.On
the map, the distance between two cities is 1 meter. What is the
actual distance between the two cities?
2. _________ The ratio of the measures of two complementary angles is 4 to 5. The smallest
measure is increased by 10%. By what percent must the larger measure be
decreased so that the two angles remain complementary?
3. _________ What is the 25th number in the pattern: 1, 2, 3, 5, 7, 10, 13, 17, 21, 26 …?
4. _________  Themeanofthesetofnumbers{87,85,80,83,84,
x
}is83.5.Whatisthemedian
of the set of six numbers? Express your answer as a decimal to the nearest tenth.
5. _________ A local ice cream shop carries vanilla ice cream, chocolate ice cream and
strawberry ice cream, as well as regular and sugar cones. Customers can
orderasingleordoublescoopoftheirfavoriteavor,ortheycanchoose
onescoopofoneavortoppedwithonescoopofanother.Ifchocolate
topped with vanilla is not the same as vanilla topped with chocolate, how
many different ice cream cone orders are possible?
6. _________  Ofthreepositiveintegers,thesecondistwicetherst,andthethirdistwicethe
second. One of these integers is 17 more than another. What is the sum of the
three integers?
7. _________ The length of a rectangular playground exceeds twice its width by 25 feet, and the
perimeter of the playground is 650 feet. What is the area of the playground?
8. _________ Theratioofboystogirlsis6:5inonerst-periodclassand3:5inasecondrst-
period class. If the school does not permit class sizes to be less than 10 students,
whatistheminimumnumberoftotalstudentsinthetworst-periodclassessothe
ratio for the combined classes is 1:1?
9. _________ Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter
of 6 inches. If the circles do not extend beyond the rectangular region, what is the
greatest possible distance between the centers of the two circles?
10. ________  Whatisthelargestprimefactorof78?
km
%
sq ft
inches
students
orders
MATHCOUNTS 2008-200930
Warm-Up 6
1. _________ The area of square I is 1 square unit. A diagonal of square I is a side of square II,
and a diagonal of square II is a side of square III. What is the area of square III?
2. _________ The heights of six students Joe, Mary, Sue, Steve, Lisa and John are 60 inches, 64
inches,58inches,68inches,63inchesand69inches.Sueis4inchesshorterthan
Joe. The girls are the three shortest students. Steve is 1 inch shorter than John.
Maryistheshorteststudent.WhatisthesumofJohn’sheightandLisa’sheight?
3. _________ The average of the three numbers
x
,
y
and
z
is equal to twice the average of
y
and
z
. What is the value of
x,
in terms of
y
and
z
?
4. _________ Giventhat−3≤
x
≤2and20
x
2
=
y
−24,whatisthesmallestpossiblevaluefor
y
?
5. _________ The current price of a pair of Lux basketball shoes is $30. The
original price had been reduced by 25%. That reduced price was
then lowered by 50% to arrive at the current price. What was
the original price?
6. _________  A2-by-4-by-8rectangularsolidispaintedred.Itiscutintounitcubesand
reassembled into a 4-by-4-by-4 cube. If the entire surface of this cube is red, how
many painted unit-cube faces are hidden in the interior of the cube?
7. _________ If a year had 364 days, then the same calendar could be used
every year by only changing the year. A “regular” year has
365 days and a leap year has 366 days. The year 2000 was a
leap year and leap years occur every 4 years between the years
2000 and 2100. Claudia has a calendar for 2009. What will be
the next year that she can use this calendar by merely changing
the year?
8. _________ A 6-question True-False test has True as the correct answer for at least
2
3
of
the questions. How many different True/False answer patterns are possible on an
answer key for this test?
9. _________ If (
m
ф
n
) =
2 2
1 1 1 1
m n
m n
+ + +
for any values of
m
and
n
,whatisthevalueof(2ф4)?
Express your answer as a common fraction.
10. ________ The new Perry Hotter book will have a cover price of $25. The local
bookstore is offering two discounts: $4.00 off and 20% off. A clever
shopper realizes that the prices will be different depending on the
order in which she claims her discounts. How much more money will
she save by taking the better-valued approach rather than the other
approach? Express your answer in cents.
Problem#4isfromthe2008ChapterTargetRound.
sq units
inches
faces
patterns
cents
$
MATHCOUNTS 2008-2009 31
Workout 3
1. _________ A car is scheduled to make a 616-mile trip in 9 hours. The car averages 60 mph
duringtherst240milesand80mphduringthenext160miles.Whatmustthe
average speed of the car be for the remainder of the trip in order for the car to
arrive on schedule?
2. _________ The front and rear wheels of a horse-drawn buggy had radii
of 14 in. and 30 in., respectively. In traveling one mile (which is
63,360 inches), what was the positive difference in the number
of revolutions made by the front and rear wheels? Express your
answer to the nearest whole number.
3. _________ Rex runs one mile in 5 minutes, Stan runs one mile in 6 minutes and Tim runs one
milein7minutes.Thereare5280feetinonemile.ByhowmanyfeetdoesTimtrail
Stan at the moment that Rex completes a one-mile run? Express your answer as a
decimal to the nearest tenth.
4. _________  Arectangularsolidboxmeasures2.75feetby4.05feetby480inches.Incubic
yards, what is the volume of the box? Express your answer as a decimal to the
nearest tenth.
5. _________ In the diagram to the right, triangle ABC is inscribed in the
circle and AC = AB. The measure of angle BAC is 42 degrees
and segment ED is tangent to the circle at point C. What is the
measure of angle ACD?
6. _________ A set of data includes all of the positive odd integers less than 100, the positive,
two-digit multiples of 10, and the numbers 4, 16 and 64. All included integers
appear exactly once in the data. What is the positive difference between the
median and the mean of the set of data? Express your answer as a decimal to the
nearest thousandth.
7. _________ Ryosuke is picking up his friend from work. The odometer on his car
reads74,568whenhepickshisfriendup,anditreads74,592when
hedropshisfriendoffathishouse.Ryosuke’scargets28milesper
gallon, and the price of one gallon of gas is $4.05. What was the cost
of the gas that was used for Ryosuke to drive his friend home from
work?
8. _________  Whatisthegreatestcommonfactorof84,112and210?
9. _________ If 10 men take 6 days to lay 1000 bricks, then how many days will it
take 20 men working at the same rate to lay 5000 bricks?
10. ________ Ayushi has six coins with a total value of 30 cents. The coins are not all the same.
TwoofAyushi’scoinswillbechosenatrandom.Whatistheprobabilitythatthe
total value of the two coins will be less than 15 cents? Express your answer as a
common fraction.
mph
revs
feet
cu yds
degrees
days
A
B
C
D
E
$
MATHCOUNTS 2008-200932
Transformations &
Coordinate Geometry
Stretch
1. _________  TriangleABChasverticeswithcoordinatesA(2,3),B(7,8)andC(−4,6).The
triangleisreectedaboutlineL.TheimagepointsareA’(2,−5),B’(7,−10)and
C’(−4,−8).WhatistheequationoflineL?
2. _________ Two vertices of an obtuse triangle are (6, 4) and (0, 0). The third
vertex is located on the negative branch of the
x
-axis. What are
the coordinates of the third vertex if the area of the triangle is
30 square units?
3. _________ The points A(2, 5), B(6, 5), C(5, 2) and D(1, 2) are the vertices of a parallelogram.
If the parallelogram is translated down two units and right three units, what will be
thecoordinatesofthenalimageofpointB?
4. _________  Namingclockwise,regularpentagonCOUNThasverticesC,O,U,N,T,respectively.
Point X is right in the center of this pentagon so that line segments from X to each
vertex create congruent triangles. If point C is rotated 144° counterclockwise
about the point X, what original vertex is the image of point C?
5. _________  IfΔABCwithverticesA(−10,2),B(−8,5),C(−6,2)isreected
over the
y
-axisandtheimageisthenreectedoverthe
x
-axis,
whatarethecoordinatesofthenalimageofpointA?
6. _________  PointsA(−4,1),B(−1,4)andC(−1,1)aretheverticesofΔABC.Whatwillbethe
coordinatesoftheimageofpointAifΔABCisrotated90°clockwiseaboutthe
origin?
7. _________ What is the angle of rotation, in degrees, about point C that
mapsthedarkerguretoitslighterimage?
8. _________  CircleThasitscenteratpointT(−2,6).CircleTisreected
across the
y
-axisandthentranslated8unitsdown.Whatarethecoordinatesof
the center of the image of circle T?
9. _________  ThepreimageofsquareABCDhasitscenterat(8,−8)andhasanareaof4square
units. The top side of the square is horizontal. The square is then dilated with the
dilation center at (0, 0) and a scale factor of 2. What are the coordinates of the
vertex of the image of square ABCD that is farthest from the origin?
10. ________  ThefourpointsA(−4,0),B(0,−4),X(0,8)andY(14,
k
) are graphed on the
Cartesian plane. If segment AB is parallel to segment XY, what is the value of
k
?
C
( , )
( , )
( , )
( , )
degrees
( , )
( , )
A
B
C
MATHCOUNTS 2008-2009 33
Problem-Solving StrategieS
NCTM’s Principles and Standards for School Mathematics recommends that the mathematics curriculum
“include numerous and varied experiences with problem solving as a method of inquiry and application.”
There are many problems within the MATHCOUNTS program that may be considered difî‚¿cult if
attacked by setting up a series of equations, but quite simple when attacked with problem-solving
strategies such as looking for a pattern, drawing a diagram, making an organized list and so on.
The problem-solving method that will be used in the following discussion consists of four basic steps:
FIND OUT
Look at the problem.
Have you seen a similar problem before?
If so, how is this problem similar? How is it different?
What facts do you have?
What do you know that is not stated in the problem?
CHOOSE A STRATEGY How have you solved similar problems in the past?
What strategies do you know?
Try a strategy that seems as if it will work.
If it doesn’t, it may lead you to one that will.
SOLVE IT Use the strategy you selected and work the problem.
LOOK BACK Reread the question.
Did you answer the question asked?
Is your answer in the correct units?
Does your answer seem reasonable?
Speciî‚¿c strategies may vary in name. Most, however, fall into these basic categories:
• Compute or Simplify (C)
• Use a Formula (F)
• Make a Model or Diagram (M)
• Make a Table, Chart or List (T)
• Guess, Check & Revise (G)
• Consider a Simpler Case (S)
• Eliminate (E)
• Look for Patterns (P)
To assist in using these problem-solving strategies, the answers to the Warm-Ups and Workouts have been
coded to indicate possible strategies. The single-letter codes above for each strategy appear in parentheses
after each answer.
In the next section, the strategies above are applied to previously published MATHCOUNTS problems.
34 MATHCOUNTS 2008-2009
MATHCOUNTS 2008-2009 35
Compute or Simplify (C)
Many problems are straightforward and require nothing more than the application of arithmetic rules.
When solving problems, simply apply the rules and remember the order of operations.
Given (6
3
)(5
4
) = (N)(900), î‚¿nd N.
FIND OUT What are we asked? The value of N that satisî‚¿es an equation.
CHOOSE A Will any particular strategy help here? Yes, factor each term in the equation into primes.
STRATEGY Then, solve the equation noting common factors on both sides of the equation.
SOLVE IT Break down the equation into each term’s prime factors.
6
3
= 6 × 6 × 6 = 2 × 2 × 2 × 3 × 3 × 3
5
4
= 5 × 5 × 5 × 5
900 = 2 × 2 × 3 × 3 × 5 × 5
Two 2s and two 3s from the factorization of 6
3
and two 5s from the factorization of 5
4
cancel the factors of 900. The equation reduces to 2 × 3 × 5 × 5 = N, so N = 150.
LOOK BACK Did we answer the question asked? Yes.
Does our answer make sense? Yes—since 900 = 30
2
= (2 × 3 × 5)
2
, we could have
eliminated two powers of 2, 3 and 5 to obtain the same answer.
Use a Formula (F)
Formulas are one of the most powerful mathematical tools at our disposal. Often, the solution to a
problem involves substituting values into a formula or selecting the proper formula to use. Some of the
formulas that will be useful for students to know are listed in the Vocabulary and Formulas section of this
book. However, other formulas will be useful to students, too. If the strategy code for a problem is (F),
then the problem can be solved with a formula. When students encounter problems for which they don’t
know an appropriate formula, they should be encouraged to discover the formula for themselves.
The formula F = 1.8C + 32 can be used to convert temperatures between degrees
Fahrenheit (F) and degrees Celsius (C). How many degrees are in the Celsius equivalent
of –22
o
F?
FIND OUT What are we trying to î‚¿nd? We want to know a temperature in degrees Celsius instead of
degrees Fahrenheit.
CHOOSE A Since we have a formula which relates Celsius and Fahrenheit temperatures, let’s
STRATEGY replace F in the formula with the value given for degrees Fahrenheit.
SOLVE IT The formula we’re given is F = 1.8C + 32. Substituting –22 for F in the equation leads to
the following solution:
–22 = 1.8C + 32
–22 – 32 = 1.8C
–30 = C
The answer is –30
o
C.
LOOK BACK Is our answer reasonable? Yes.
36 MATHCOUNTS 2008-2009
Make a Model or Diagram (M)
Mathematics is a way of modeling the real world. A mathematical model has traditionally been a form
of an equation. The use of physical models is often useful in solving problems. There may be several
models appropriate for a given problem. The choice of a particular model is often related to the student’s
previous knowledge and problem-solving experience. Objects and drawings can help to visualize problem
situations. Acting out the situation also is a way to visualize the problem. Writing an equation is an
abstract way of modeling a problem situation. The use of modeling provides a method for organizing
information that could lead to the selection of another problem-solving strategy.
Use Physical Models
Four holes are drilled in a straight line in a rectangular steel plate. The distance between
hole 1 and hole 4 is 35 mm. The distance between hole 2 and hole 3 is twice the distance
between hole 1 and hole 2. The distance between hole 3 and hole 4 is the same as the
distance between hole 2 and hole 3. What is the distance, in millimeters, between the
center of hole 1 and the center of hole 3?
FIND OUT We want to know the distance between hole 1 and hole 3.
What is the distance from hole 1 to hole 4? 35 mm
What is the distance from hole 1 to hole 2? Half the distance from hole 2 to hole 3.
What is the distance from hole 3 to hole 4? The same as from hole 2 to hole 3.
CHOOSE A Make a model of the problem to determine the distances involved.
STRATEGY
SOLVE IT Mark off a distance of 35 mm.
Place a marker labeled #1 at the zero point and one labeled #4 at the 35-mm point.
Place markers #2 and #3 between #1 and #4.
1) Move #2 and #3 until the distances between #2 & #3 and #3 & #4 are equal.
2) Is the distance between #1 & #2 equal to half the distance between #2 & #3?
Adjust the markers until both of these conditions are met.
Measure the distances to double check. The distance between #1 and #3 is 21 mm.
LOOK BACK Does our answer seem reasonable? Yes, the answer must be less than 35.
Act Out the Problem
There may be times when you experience difî‚¿culty in visualizing a problem or the procedure necessary
for its solution. In such cases you may î‚¿nd it helpful to physically act out the problem situation. Y
ou
might use people or objects exactly as described in the problem, or you might use items that represent
the people or objects. Acting out the problem may itself lead you to the answer, or it may lead you to
î‚¿nd another strategy that will help you î‚¿nd the answer. Acting out the problem is a strategy that is very
effective for young children.
There are ve people in a room, and each person shakes every other person’s hand
exactly one time. How many handshakes will there be?
FIND OUT We are asked to determine the total number of handshakes.
How many people are there? 5
How many times does each person shake another’s hand? Only once.
CHOOSE A Would it be possible to model this situation in some way? Yes, pick î‚¿ve friends and ask
STRATEGY them to act out the problem.
MATHCOUNTS 2008-2009 37
Should we do anything else? Keep track of the handshakes with a list.
SOLVE IT Get î‚¿ve friends to help with this problem.
Make a list with each person’s name at the top of a column.
Have the rst person shake everyone’s hand. How many handshakes were there? Four.
Repeat this four more times with the rest of the friends. Write down who each person
shook hands with. Our table should look something like this:
There were a total of 20 handshakes. But notice that each person actually shook everyone
else’s hand twice. (For example, Rhonda shook Jagraj’s hand, and Jagraj shook Rhonda’s
hand.) Divide the total number of handshakes by two to î‚¿nd out the total number if each
person had shaken every other person’s hand only once. There were 10 handshakes.
LOOK BACK Did we answer the question? Yes.
Does our answer seem reasonable? Yes.
Use Drawings or Sketches
If an eight-inch-square cake serves four people, how many 12-inch-square cakes are
needed to provide equivalent servings to 18 people?
FIND OUT We are to nd how many 12 × 12 cakes are needed.
How big is the original cake? 8 × 8
How many people did it feed? 4
How big are the other cakes? 12 × 12
How many people must they feed? 18
CHOOSE A How should we approach this problem? Diagram the cakes to understand the size
STRATEGY of the portions.
SOLVE IT Draw an 8 × 8 cake and cut it into 4 equal pieces. Since each piece is a square with side
length of 4, the area of each piece is 4 × 4 = 16 square inches.
So each person gets 16 square inches of cake.
Rhonda Jagraj Rosario Kiran Margot
Jagraj Rosario Kiran Margot Rhonda
Rosario Kiran Margot Rhonda Jagraj
Kiran Margot Rhonda Jagraj Rosario
Margot Rhonda Jagraj Rosario Kiran
4
4
38 MATHCOUNTS 2008-2009
18 people times 16 square inches per person equals 288 total square inches of cake
needed.
We know that a 12 × 12 cake contains 144 square inches of cake.
288 divided by 144 equals 2, so two 12 × 12 cakes are required to feed 18 people.
LOOK BACK Did we answer the correct question, and does our answer seem reasonable? Yes.
Use Equations
Lindsey has a total of $82.00, consisting of an equal number of pennies, nickels, dimes
and quarters. How many coins does she have in all?
FIND OUT We want to know how many coins Lindsey has.
How much money does she have total? $82.00
How many of each coin does she have? We don’t know exactly, but we know that she has
an equal number of each coin.
CHOOSE A We know how much each coin is worth, and we know how much all of her coins are
STRATEGY worth total, so we can write an equation that models the situation.
SOLVE IT Let p be the number of pennies, n the number of nickels, d the number of dimes and q the
number of quarters.
We then have the equation p + 5n + 10d + 25q = 8200.
We know that she has an equal number of each coin, so p = n = d = q. Substituting p for
the other variables gives an equation in just one variable. The equation above becomes
p
+ 5p + 10p + 25p = 41p = 8200, so p = 200.
Lindsey has 200 pennies. Since she has an equal number of each coin, she also has
200 nickels, 200 dimes and 200 quarters. Therefore, she has 800 coins.
LOOK BACK Did we answer the question asked? Yes.
Does our answer seem reasonable? Yes, we know the answer must be less than 8200 (the
number of coins if they were all pennies) and greater than 328 (the number of coins if
they were all quarters).
Make a Table, Chart or List (T)
Making a table, chart, graph or list is a way to organize data presented in a problem. This problem-solving
strategy allows the problem solver to discover relationships and patterns among data.
Use Tree Diagrams or Organized Lists
Customers at a particular yogurt shop may select one of three avors of yogurt. They
may choose one of four toppings. How many one‑avor, one‑topping combinations are
possible?
FIND OUT What question do we have to answer? How many avor‑topping combinations are
possible?
How many avors are available? 3
How many toppings are available? 4
MATHCOUNTS 2008-2009 39
Are you allowed to have more than one avor or topping? No, the combinations must
have only one avor and one topping.
CHOOSE A How could we organize the possible combinations to help? With letters and numbers in a
STRATEGY list.
SOLVE IT Make an organized list. Use F and T to denote either avor or topping. Use the numbers
1 – 3 and 1–4 to mark different avors and toppings.
F1T1, F1T2, F1T3, F1T4
F2T1, F2T2, F2T3, F2T4
F3T1, F3T2, F3T3, F3T4
Now count the number of combinations. There are 12 combinations possible.
LOOK BACK Did we answer the question asked? Yes.
Does our answer seem reasonable? Yes.
make a Chart
How many hours will a car traveling at 45 miles per hour take to catch up with a car
traveling at 30 miles per hour if the slower car starts one hour before the faster car?
FIND OUT What is the question we have to answer? How long does it take for the
faster car to catch the slower car?
What is the speed of the slower car? 30 miles per hour
What is the speed of the faster car? 45 miles per hour
CHOOSE A What strategy will help here? We could model this on paper, but accuracy
STRATEGY would suffer. We also could use equations. But let’s make a table with the time and
distance traveled since that will explicitly show what’s happening here.
SOLVE IT Make a table with two rows and four columns.
The rows will identify the cars, and the columns will mark the hours.
Where the rows and columns intersect will indicate distance traveled since distance
equals the speed times the amount of time traveled.
At the end of the î‚¿rst hour, the faster car was just starting. At the end of the second hour,
the faster car had gone 45 miles. At the end of the third hour, the faster car had gone 90
miles. This equals the distance traveled by the slower car in three hours. So, the faster car
traveled for only two hours.
LOOK BACK Did we answer the question asked? Yes.
Does our answer seem reasonable? Yes.
Hour
Car
1 2 3 4
Slow Car 30 60 90 120
Fast Car 0 45 90 135
40 MATHCOUNTS 2008-2009
guess, Check & revise (g)
The Guess, Check & Revise strategy for problem solving can be helpful for many types of problems.
When using this strategy, students are encouraged to make a reasonable guess, check the guess and
revise the guess if necessary. By repeating this process a student can arrive at a correct answer that has
been checked. Using this strategy does not always yield a correct solution immediately, but it provides
information that can be used to better understand the problem and may suggest the use of another strategy.
Students have a natural afî‚¿nity for this strategy and should be encouraged to use it when appropriate.
To use the Guess, Check & Revise strategy, follow these steps:
1. Make a guess at the answer.
2. Check your guess. Does it satisfy the problem?
3. Use the information obtained in checking to help you make a new guess.
4. Continue the procedure until you get the correct answer.
Leah has $4.05 in dimes and quarters. If she has î‚¿ve more quarters than dimes, how
many of each does she have?
FIND OUT What are we asked to determine? We need to î‚¿nd how many dimes and how many
quarters Leah has.
What is the total amount of money? $4.05
What else do we know? There are î‚¿ve more quarters than dimes.
CHOOSE A Will listing combinations help? Yes, but creating an extended list of possible
STRATEGY combinations of dimes and quarters could be cumbersome to create.
What other strategy would work? Pick a number, try it and adjust the estimate.
SOLVE IT Try 5 dimes. That would mean 10 quarters.
5 × $0.10 + 10 × $0.25 = $3.00
Increase the number of dimes to 7.
7 × $0.10 + 12 × $0.25 = $3.70
Try again. This time use 8 dimes.
8 × $0.10 + 13 × $0.25 = $4.05
Leah has 8 dimes and 13 quarters.
LOOK BACK Did we answer the question asked, and does our answer seem reasonable? Yes.
Trevor had 60 markers he could turn in at the end of the year for extra-credit points he
had earned during the year. Some markers were worth one point and others were worth
two points. If he was entitled to a total of 83 extra-credit points, how many one-point
markers did he have?
FIND OUT What question are we trying to answer? The question is how many one-point markers did
Trevor have.
What is the total number of markers he had? 60.
What were their possible values? One or two points.
What was the total value of all the markers? The markers totaled 83 points.
CHOOSE A How can we approach this problem? Make a table of the possible number of
STRATEGY markers and their total value.
SOLVE IT Make a guess as to the î‚¿rst value. We can adjust our guess as we get closer to the desired
answer.
Pick 10 as the number of one-point markers. This means he has 50 two-point markers
since we know he has 60 markers total. The value of this combination is 110 points.
We can keep track of our guesses in a table by listing the number of one-point markers,
the number of two-point markers and the total number of points various combinations
would give.
MATHCOUNTS 2008-2009 41
# of 1-point Markers # of 2-point Markers Total Value
10 50 110
50 10 70
40 20 80
38 22 82
37 23 83
Trevor had 37 one-point markers.
LOOK BACK Did we answer the question? Yes.
Does our answer seem reasonable? Yes, we know the answer has to be less than 60. Also,
23 points more than 60 implies that 23 markers were worth 2 points.
Consider a Simpler Case (S)
The problem-solving strategy of simplifying most often is used in conjunction with other strategies.
Writing a simpler problem is one way of simplifying the problem-solving process. Rewording the
problem, using smaller numbers or using a more familiar problem setting may lead to an understanding of
the solution strategy to be used. Many problems may be divided into simpler problems to be combined to
yield a solution. Some problems can be made simpler by working backwards.
Sometimes a problem is too complex to solve in one step. When this happens, it is often useful to simplify
the problem by dividing it into cases and solving each one separately.
Divide Into Smaller Problems
Three shapes—a circle, a rectangle and a square—have the same area. Which shape has
the smallest perimeter?
FIND OUT We want to know which of three shapes has the smallest perimeter.
CHOOSE A Will any particular strategy help here? Yes, we can compare the perimeters of the
STRATEGY shapes pairwise. This will be easier than calculating the area of each since numbers are
not given.
SOLVE IT First, compare the circumference of the circle to the perimeter of the square. They have
equal area, so the area of the circle, πr
2
, equals the area of the square, s
2
. Consequently,
the perimeter of the square will be slightly greater than the circumference of the circle.
Next, compare the perimeter of the square to the perimeter of the rectangle. A square is a
quadrilateral which has minimum perimeter, so the perimeter of the square must be less
than the perimeter of the rectangle.
By the transitive property, then, the perimeter of the rectangle will be greater than the
circumference of the circle. Hence, the circle has the smallest perimeter.
LOOK BACK Did we answer the question asked? Yes.
Does our answer make sense? Yes. If we arbitrarily choose 100 units
2
as the area of each
shape, the circumference of the circle is roughly 35.5 units, the perimeter of the square
is 40 units, and the perimeter of the rectangle could be any amount greater than 40 units
and less than 100 units.
42 MATHCOUNTS 2008-2009
Work Backwards
A student needs at least a 95% average to receive a grade of A. On the î‚¿rst three tests the
student averaged 92%. What is the minimum the student must average on the last two
tests to receive a grade of A?
FIND OUT We are asked to î‚¿nd what a student must average on her last two tests to get an A.
What average is required for an A? 95%
How many tests will be î‚¿gured into the average? 5
How many test has she taken so far? 3
What is her average on the î‚¿rst three tests? 92%
CHOOSE A What strategy would work well in this situation? Work backwards from the minimum
STRATEGY required average needed for an A to î‚¿nd the scores needed on the last two tests.
SOLVE IT Work backwards from the required average on all î‚¿ve tests.
The average of the tests must be 95%. There are î‚¿ve tests so the total number of points
scored on the ve tests must be, at least, 5 × 95 = 475.
So far, the average is 92% on three tests. While we don’t know all of the individual
scores, the total number of points scored on the three tests must be 3 × 92 = 276.
475 points required minus 276 scored so far equals 199 required on the next two tests.
199 divided by 2 equals 99.5.
The student must average 99.5% on her next two tests if she is to get an A.
LOOK BACK Did we answer the question asked? Yes.
Does our answer seem reasonable? Yes, we knew we were looking for a number between
95 and 100.
Eliminate (E)
The strategy of elimination is commonly used by people in everyday life. In a problem-solving context,
students must list and then eliminate possible solutions based upon information presented in the problem.
The act of selecting a problem-solving strategy is an example of the elimination process. Logical
reasoning is a problem-solving strategy that is used in all problem-solving situations. It can result in
the elimination of incorrect answers, particularly in “if-then” situations and in problems with a listable
number of possible solutions.
What is the largest two-digit number that is divisible by 3 whose digits differ by 2?
FIND OUT What are we asked to î‚¿nd? A certain number.
What do we know about the number? The number is less than 100. It is divisible by 3.
The digits of the number differ by 2.
CHOOSE A What strategy will help here? Working backwards from 99, list numbers and
STRATEGY eliminate those that do not satisfy the conditions given. (Notice that we have already
eliminated numbers greater than 99.)
SOLVE IT 99, 98, 97, 96, 95, 94, 93, 92, 91, 90,
89, 88, 87, 86, 85, 84, 83, 82, 81, 80,
79, 78, 77, 76, 75, 74, 73, 72, 71, 70, . . .
Eliminate those numbers that are not divisible by 3:
MATHCOUNTS 2008-2009 43
99, 98, 97, 96, 95, 94, 93, 92, 91, 90,
89, 88, 87, 86, 85, 84, 83, 82, 81, 80,
79, 78, 77, 76, 75, 74, 73, 72, 71, 70, . . .
From these, eliminate all numbers whose digits do not differ by 2:
99, 96, 93, 90, 87, 84, 81, 78, 75, 72, . . .
75 is the largest number that remains.
LOOK BACK Did we answer the question asked? Yes.
Do we have a two-digit number divisible by 3 whose digits differ by 2? Yes.
look for Patterns (P)
When students use this problem-solving strategy, they are required to analyze patterns in data and make
predictions and generalizations based on their analysis. They then must check the generalization
against the information in the problem and possibly make a prediction from, or extension of, the given
information. A pattern is a regular, systematic repetition. A pattern may be numerical, visual or behavioral.
By identifying the pattern, you can predict what will come next and what will happen again and again in
the same way. Looking for patterns is a very important strategy for problem solving and is used to solve
many different kinds of problems. Sometimes you can solve a problem just by recognizing a pattern, but
often you will have to extend a pattern to î‚¿nd a solution. Making a number table often reveals patterns,
and for this reason it is frequently used in conjunction with this strategy.
Laura was given an ant farm by her grandparents for her 13th birthday. The farm could
hold a total of 100,000 ants. Laura’s farm had 1500 ants when it was given to her. If the
number of ants in the farm on the day after her birthday was 3000 and the number of ants
the day after that was 6000, in how many days will the farm be full?
FIND OUT We need to know when the ant farm will be full.
How many ants will the farm hold? 100,000
How many ants are in the farm the î‚¿rst day? 1500
How many ants are in the farm the second day? 3000
How many ants are in the farm the third day? 6000
CHOOSE A Is a pattern developing? Yes, each day twice as many ants are in the farm as
STRATEGY the day before. Make a table to count the ants systematically.
SOLVE IT Draw a table with two lines for numbers.
The top line is the number of days after Laura’s birthday, and the bottom line is the
number of ants in the farm on that day.
# days 0 1 2 3 4 5 6 7
# ants 1500 3000 6000 12,000 24,000 48,000 96,000 192,000
The ant farm will be full seven days after her birthday.
LOOK BACK Read the question again. Did we answer all of the question? Yes.
Does our answer seem reasonable? Yes.
What assumption are we making? We are assuming that the pattern—the number of ants
doubles each day—continues indenitely.
44 MATHCOUNTS 2008-2009
MATHCOUNTS 2008-2009 45
voCabUlary anD FormUlaS
The following list is representative of terminology used in the problems but should not be viewed as
all-inclusive. It is recommended that coaches review this list with their Mathletes.
intersection
inverse variation
irrational number
isosceles
LCM
lateral surface area
lateral edge
lattice point(s)
linear equation
mean
median of a set of data
median of a triangle
midpoint
mixed number
mode(s) of a set of data
multiple
multiplicative inverse
(reciprocal)
natural number
numerator
obtuse angle
octagon
odds (probability)
octahedron
opposite of a number (additive
inverse)
ordered pair
ordinate
origin
palindrome
parallel
parallelogram
Pascal’s triangle
pentagon
percent increase/decrease
perimeter
permutation
perpendicular
planar
polygon
polyhedron
prime factorization
prime number
principal square root
prism
degree measure
denominator
diagonal of a polygon
diagonal of a polyhedron
diameter
difference
digit
digit-sum
direct variation
dividend
divisible
divisor
edge
endpoint
equation
equiangular
equidistant
equilateral
evaluate
expected value
exponent
expression
exterior angle of a polygon
factor
factorial
Fibonacci sequence
î‚¿nite
formula
frequency distribution
frustum
function
GCF
geometric sequence
height (altitude)
hemisphere
hexagon
hypotenuse
image(s) of a point(s) (under a
transformation)
improper fraction
inequality
inî‚¿nite series
inscribe
integer
interior angle of a polygon
abscissa
absolute value
acute angle
additive inverse (opposite)
adjacent angles
algorithm
alternate interior angles
alternate exterior angles
altitude (height)
area
arithmetic mean
arithmetic sequence
base 10
binary
bisect
box-and-whisker plot
center
chord
circle
circumscribe
circumference
coefî‚¿cient
collinear
combination
common divisor
common denominator
common factor
common fraction
common multiple
complementary angles
composite number
compound interest
concentric
cone
congruent
convex
coordinate plane/system
coordinates of a point
corresponding angles
counting numbers
counting principle
cube
cylinder
data
decimal
46 MATHCOUNTS 2008-2009
probability
product
proper divisor
proper factor
proper fraction
proportion
pyramid
Pythagorean Triple
quadrant
quadrilateral
quotient
radius
random
range of a data set
rate
ratio
rational number
ray
real number
reciprocal (multiplicative
inverse)
rectangle
reection
regular polygon
relatively prime
remainder
repeating decimal
revolution
rhombus
right angle
right circular cone
right circular cylinder
right polyhedron
right triangle
rotation
scalene triangle
scientiî‚¿c notation
segment of a line
semicircle
sequence
set
similar î‚¿gures
simple interest
slope
slope-intercept form
solution set
sphere
square
square root
stem-and-leaf plot
sum
supplementary angles
system of equations/
inequalities
tangent î‚¿gures
tangent line
term
terminating decimal
tetrahedron
total surface area
transformation
translation
trapezoid
triangle
triangular numbers
trisect
union
unit fraction
variable
vertical angles
vertex
volume
whole number
x-axis
x-coordinate
x-intercept
y-axis
y-coordinate
y-intercept
The list of formulas below is representative of those needed to solve MATHCOUNTS problems but
should not be viewed as the only formulas that may be used. Many other formulas that are useful in
problem solving should be discovered and derived by Mathletes.
CIRCUMFERENCE
Circle C = 2 × � × r = � × d
AREA
Square A = s
2
Rectangle A = l × w = b × h
Parallelogram A = b × h
Trapezoid A =
1
2
(b
1
+ b
2
) × h
Circle A = π × r
2
Triangle A =
1
2
× b × h
Triangle
( )( )( )= − − −A s s a s b s c
Equilateral triangle
2
3
4
=
s
A
Rhombus A =
1
2
× d
1
× d
2
SURFACE AREA & VOLUME
Sphere SA = 4 × π × r
2
Sphere V =
4
3
× π × r
3
Rectangular prism V = l × w × h
Circular cylinder V = π × r
2
× h
Circular cone V =
1
3
× π × r
2
× h
Pyramid V =
1
3
× B × h
Pythagorean Theorem c
2
= a
2
+ b
2
Counting/
Combinations
with semi-perimeter s and sides a, b and c
!
( !)(( )!)
=
−
n r
n
C
r n r
MATHCOUNTS 2008-2009 47
REFERENCES
Problem-Solving References
Dolan, Daniel T. and James Williamson. Teaching Problem-Solving Strategies. Reading, MA: Scott-Foresman Addison-Wesley
Publishing Co., 1983, 1st ed.
Goodnow, J. et al. The Problem Solver (Binders). Mountain View, CA: Creative Publications, 1987 & 1988.
Krulik, Stephen, ed. Problem Solving in School Mathematics. 1980 Yearbook of the National Council of Teachers of
Mathematics. Reston, VA: NCTM, 1980.
The Lane County Mathematics Project. Problem Solving in Mathematics Grades 4-9. Palo Alto, CA: Dale Seymour Publications,
1983.
Lenchner, George. Creative Problem Solving in School Mathematics. Boston, MA: Houghton Mifin McDougal Littell Co.,
1983.
Polya, George. How To Solve It. Princeton, NJ: Princeton University Press, 1988.
Seymour, Dale.
Favorite Problems, Grades 5-7. Palo Alto, CA: Dale Seymour Publications, 1982.
48 MATHCOUNTS 2008-2009
MATHCOUNTS 2008-2009 49
1. 22.20 (C)
2. 19 (F, M)
3. 42 (C)
4. 6 (P, T)
Warm-Up 1
5. 90 (F, P)
6. −80 (F,G)
7. 131 (P, T)
8. 54 (P,T)
9. 52 (C,G)
10. 120 (F, T)
Answers
Warm-Up 2
1. 38 (F,P)
2. 135 (F, M)
3. 144 (C)
4.
1
6
(C)
5. 74 (C, M)
6. 40 (F,G)
7. 4 (F,G,M)
8.
1
25,000
(C, F)
9. 10 (F,G)
10. 41 (C, M)
Answers
Workout 1
1. 193 (C,G,P)
2. 40 (C, M)
3. 575 (C, F)
4. 125 (C, F)
5. 60 (C, F)
6. 9 (C)
7. 8 (C,P)
8. 188 (F,G,M)
9.
8
9
(C, M, P)
10.
1
50
(P, S)
Answers
1. 160 (C,F,G)
2. 15 (C,F,G,M,P,T)
3. 48.75 (C)
4.
1
3
13
 (C,F,G)
Warm-Up 3
5. 1* (C, F, M)
6.
1
840
(C, F, M)
7. 380 (C,G,M,S,T)
8. 63 (C,M,S)
9. 47 (C, M, T)
10. 108 (C,F,P)
Answers
*
The plural form of the units will always be provided in the answer blank even if the answer appears
to require the singular form of units.
ANSWERS TO HANDBOOK PROBLEMS
MATHCOUNTS 2008-200950
Answers
Answers
Warm-Up 4
1. 832 (C,F,M)
2. 3 (C, F)
3. 810 (C,P,T)
4. 30 (C)
5. 24 (P, T)
6. 30 (C, P)
7. 264 (C, F, M)
8.
1
1
5
(C, F)
9. 10 (F, M)
10. 38 (C,F,G,M,S,T)
Workout 2
1. 21 (C,E,G,P,T)
2. 18.4 (C,G,M,S,T)
3. (7,2,1) (E,G,T)
4. 97 (G)
5. 523.6 (C, F, M)
6. 2.5 (C, F, M, S)
7. 841 (C,E,F,G,P,T)
8.
29
32
(C, F, M, S, T)
9. 183 (C,E,F,G)
10. 29 (C)
1. 125 (C, M, T)
2. 8 (C,F,M)
3. 157 (C, F, P, T)
4. 83.5 (C,F)
Warm-Up 5
5. 24 (C, F, M, T)
6. 119 (C,E,G,P)
7. 22,500 (C, F, M)
8. 60 (G,T)
9. 10 (C ,F, M)
10. 13 (C,E,G)
Answers
MATHCOUNTS 2008-2009 51
Warm-Up 6
1. 4 (C, F, M)
2. 132 (E,G,T)
3. 2
y
+ 2
z
or 2(
y
+
z
) (F)
4. 24 (C, F, P)
5. 80or80.00 (C,F)
6. 16 (C ,F, M)
7. 2015 (P, T)
8. 22 (C,F,T)
9.
17
16
(C, F)
10. 80 (C,M)
Answers
Answers
Workout 3
1. 72 (C, F, T)
2. 384 (C,F)
3. 628.6 (C,F)
4. 16.5 (C, F)
5. 69 (F)
6. 0.565 or .565 (C, P)
7. 3.47 (C, F)
8. 14 (C,G)
9. 15 (C, M)
10.
2
3
(C, M, T)
Transformations & Coordinate Geometry
Stretch
1.
y
=−1 (M,P)
2. (−15,0) (C,F,M)
3. (9, 3) (F)
4. N (F, M)
5. (10, –2) (F, M)
6. (1, 4) (M)
7. 180 (G,M)
8. (2,−2) (F,M)
9. (18,−18) (F,M)
10. −6 (F,M)
Answers
MATHCOUNTS 2008-200952
MATHCOUNTS 2008-2009 53
SOLUTIONS TO HANDBOOK PROBLEMS
The solutions provided here are only possible solutions. It is very likely that you and/or your students will come up
with additional—and perhaps more elegant—solutions. Happy solving!
Warm-Up 1
Problem 1. If 3 dozen tomatoes cost $6.66, then 1 dozen cost $6.66 ÷ 3 = $2.22 and 10 dozen must cost $2.22 × 10 = $22.20.
Problem 2. The square with a perimeter of 40 inches must have a side length of 40 ÷ 4 = 10 inches, and an area of 10
2
= 100 square inches. The
square with a perimeter of 36 inches must have a side length of 36 ÷ 4 = 9 inches, and an area of 9
2
= 81 square inches. The positive difference
between these two areas is 100 – 81 = 19 square inches.
Problem 3. The prime numbers on a standard six-sided die are 2, 3 and 5. These numbers occurred a total of 5 + 9 + 7 = 21 times out of the
50 rolls, which is 21 ÷ 50 = 0.42 or 42% of the rolls.
Problem 4. We can think of 1000 as 20 × 50. The factors of 50 are 1, 2, 5, 10, 25 and 50. If we multiply each of these 6 factors of 50 by 20, we
will get the six (6) factors of 1000 that can be divided evenly by 20. They are 20, 40, 100, 200, 500 and 1000.
Problem 5. Let’s call the side length of the square s. This makes the perimeter of the square 4s, which we know is 144 units. Solving 4s = 144 for
s, we get s = 36. We also can say that the perimeter of each rectangle is 2(s + 0.25s). Since we found that s = 36, we know that the perimeter of
each rectangle is 2(36 + (0.25)(36)) = 90 units.
Problem 6. Suppose we add 18 to the lesser of the two integers. Then they would have the same value and their sum would be 2 + 18 = 20. The
larger integer must be 20 ÷ 2 = 10, and the smaller integer must be 10 – 18 = – 8. The product of the two integers is 10 × −8 = −80.
Problem 7. The median of a set of consecutive integers is the middle value of that set. Since the median is an even number, but there are only odd
integers in this set, there must be an even number of integers in the set. The set must be {131, 133, 135, 137, 139, 141, 143, 145}, and 131 is the
least integer in the set.
Problem 8. In the 100s, we have 108, 118, 128, etc., and we have all the numbers in the 180s, but we must exclude 188 since it has two 8s. That
means there are 18 integers with exactly one digit that is an 8. There are also 18 integers in the 200s and the 300s, so there are 3 × 18 = 54 total.
Problem 9. Since Dan played more games than Ben, Ben could have played at most 13 games. Since Ben played exactly 1 out of 4 games, the
maximum number of games the team could have played is 4 × 13 = 52 games.
Problem 10. Suppose Peter rode x miles on the î‚¿rst day. Then he rode x + 10 miles on the second day, x + 20 miles on the third day, x + 30 miles
on the fourth day, and x + 40 miles on the î‚¿fth day. We know that Peter rode 5x + 100 miles, so 5x + 100 = 500. Solving for x, we get 5x = 400
and then x = 80 miles. He must have ridden 80 + 40 = 120 miles on the î‚¿fth day. Another solution is to see that Peter would have ridden 100
miles each day if he rode the same distance each day. Keeping the middle day (the third day) at 100 and adding/subtracting 10 miles from the
other days according to the problem, we see he rode 80, 90, 100, 110 and 120 on the î‚¿ve days.
Warm-Up 2
Problem 1. The î‚¿rst triangle requires three matchsticks. After that, it takes only two matchsticks to make a new triangle in the pattern. Thus, the
number of matchsticks, m, is one more than twice the number of triangles, t. If 77 matchsticks are used, we can solve the equation 77 = 2t + 1.
There must be (77 – 1) ÷ 2 = 76 ÷ 2 = 38 triangles.
Problem 2. If the perimeter of the room is 48 feet, then the semiperimeter is half that or 24 feet. This is the sum of the length and the width. The
part-to-part ratio 5:3 is a total of 8 parts, so each part must be worth 24 ÷ 8 = 3 feet. That means the length is 5 × 3 = 15 feet and the width is
3 × 3 = 9 feet, so the area must be 15 × 9 = 135 square feet.
Problem 3. We don’t need to know that 7! = 5040 to nd the greatest perfect square factor of 7!. We should look at the prime factorization of 7!,
which is 7 × 6 × 5 × 4 × 3 × 2 × 1 = 7 × (2 × 3) × 5 × (2 × 2) × 3 × 2 × 1. We have four factors of 2 and two factors of 3, so the greatest perfect
square factor of 7! is 2
4
× 3
2
= 16 × 9 = 144.
Problem 4. There are two numbers less than 3 on a standard die (namely, 1 and 2), so there is a 2/6 chance of rolling one of those. There is also a
1/2 chance of tossing a tail on the dime. The probability that these two things will happen together is 2/6 × 1/2 = 1/6.
Problem 5. If we add the number of students who have dogs to the number of students who have cats, we have double counted the students who
have both dogs and cats. Thus, there must be 152 + 120 – 46 = 226 students who have at least one of these pets. That leaves 300 – 226 =
74 students in the survey who did not have either a cat or a dog.
Problem 6. If Alicia’s average score on her ve tests is 88 points, then the sum of her scores must be 88 × 5 = 440 points. If she earned
100 points on four of the tests, then she could have earned a score as low as 40 points on the other test.
Problem 7. The area is the square of the side length and the perimeter is 4 times the side length. If s
2
= 4s, then the side length, s, is 4 units.
MATHCOUNTS 2008-200954
Problem 8. One kilometer is 1000 meters and 1 meter is 100 centimeters, so 1 km = 100,000 cm. The ratio 4 cm to 1 km becomes 4 cm to
100,000 cm, which written as a common fraction, is 1/25,000.
Problem 9. If x is Sue’s age now, x + 30 = 4x. When we solve for x we nd that Sue is now 10 years old.
Problem 10.The area of the legal-sized piece of paper is 8.5 × 14 = 119 square inches. When a one-inch border is cut from all four sides, we are left
with a paper that measures 6.5 inches by 12 inches and has an area of 6.5 × 12 = 78 square inches. The difference 119 – 78 = 41 is the number of
square inches that have been cut off.
Workout 1
Problem 1. The two facing pages will differ by 1, so each is close to the square root of 9312, which is about 96.4987. The two pages must be 96 and
97, and their sum is 193. Alternately, by looking at the prime factorization of 9312, which is 2
5
× 3 × 97, we see that 96 × 97 = 9312. So, again,
96 + 97 = 193.
Problem 2. We are dealing in both meters and feet in this problem, which can be confusing. A careful reading, however, reveals that the 9 meters
that Henry walked due north are later eliminated by the 9 meters that he walked due south. At the end, Henry is 24 feet east and 32 feet south of his
original location. These are the two legs of a right triangle, so we can î‚¿gure out the length of the hypotenuse of the triangle using the Pythagorean
Theorem. Actually, 24 is 3 × 8 and 32 is 4 × 8, so this is just a multiple of the 3-4-5 triangle. The hypotenuse—and Henry’s distance from his starting
point—must be 5 × 8 = 40 feet.
Problem 3. If we knew last year’s enrollment at Liberty Middle School, we would multiply by 1.04 to get the new enrollment of 598 students.
Working backward, we can divide 598 by 1.04 to get 575 students. Alternatively, we could solve the equation x + 0.04x = 598, where x is last year’s
enrollment.
Problem 4. John will make $70 – $30 = $40 on each lamp that he sells. He will need to sell $5000 ÷ $40 = 125 lamps to recover his set-up cost.
Problem 5. Using the Pythagorean Theorem, we calculate that the other leg of the original right triangle must be √(29
2
– 21
2
) = √(841 – 441) =
√400 = 20 inches. Since 87 is 3 times 29, the length of the shortest side of the second triangle must be 3 × 20 = 60 inches.
Problem 6. Substituting 4 and –1 into the rule, we get (2 × 4 + (−1)
2
) = (8 + 1) = 9.
Problem 7. When we add consecutive counting numbers starting with 1, we get numbers known as triangular numbers. We are essentially looking for
a triangular number that is also a square number. The rst few triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36… We see that when n = 8, we get the
number 36, which is both a triangular number and a square number, thus n = 8.
Problem 8. For this problem we must remember the Triangle Inequality Theorem that states that the shortest side must be longer than the positive
difference of the other two sides. We will try to make a long skinny triangle with side AB as short as possible. First we try making AB equal to 1 unit.
Then the other two sides must have a difference less than 1 unit in order to form a triangle. The closest we can come with integers is 191 and 192,
but that won’t work. The shorter sides will lay at on the longest side and will fail to make a triangle. Next we try making AB equal to 2 units. If the
other two sides were 191 each, we would have a triangle, but all three sides would not have different lengths. If the other two sides were 190 and 192,
we wouldn’t have a triangle. Finally, we try making AB equal to 3 units. Then the other two sides could be 190 and 191 units, and we can now form a
triangle. The greatest possible difference is therefore 191 – 3 = 188 units.
Problem 9. Since Sue’s line is steeper than Mary’s, the slope must be closer to 1 than Mary’s slope. The only fraction that has a numerator and a
denominator that are both single digits and accomplishes this is 8/9. We know to look at ninths because they are smaller than eighths, which will
allow us to get closer to 1.
Problem 10. After the subtractions are performed, each fraction in the pattern has a numerator that is one less than its denominator. The product then
reduces quite nicely, leaving just the rst numerator and the last denominator, as follows: 1/2 × 2/3 × 3/4 × … × 49/50 = 1/50.
Warm-Up 3
Problem 1. If we let the measure of angle B equal x, then the measure of angle A is 8x. Since angles A and B are supplementary, we can say that
x + 8x = 180. If we solve for x we î‚¿nd that x = 20. Thus, angle A = 8(20) = 160 degrees.
Problem 2. Rewriting the statement “51 is 3 less than twice some number” in algebra, we get 51 = 2x – 3. Now we solve for x, and nd that
54 = 2x and x = 27. This is the second number. (Notice that we added three to 51 and divided by 2 to get this result.) Repeating the process, we write
27 = 2y – 3 and then solve for y. We get 30 = 2y and y = 15. The rst number in the sequence is 15.
Problem 3. To price an item at 25% over the wholesale cost, we can multiply by 1.25. In this case, we get $39 × 1.25 = $48.75.
Problem 4. We need to realize that Rashid spent more time traveling at the slower speed even though the distances are the same. Let’s convert each
speed to miles per hour: 2 miles every 15 minutes is 8 miles per hour, and 2 miles every 3 minutes is 40 miles per hour. Then Rashid would have
taken 40 ÷ 8 = 5 hours to get to Pythagoras and 40 ÷ 40 = 1 hour to get back. That’s a total of 6 hours to go 80 miles, which is 80/6 = 13
1
/
3
miles per
hour.
Problem 5. If the area of square ABCD is 9 square units, then the side length of square ABCD is 3 units. The side length of square PQRS must be
1 unit.
MATHCOUNTS 2008-2009 55
Problem 6. The probability that Stu pulls the M out î‚¿rst is 1/7. The probability that he then pulls the A out is 1/6. The probability that he pulls out M,
A, T, H in this order is 1/7 × 1/6 × 1/5 × 1/4 = 1/840.
Problem 7. We should make x + y = 39, and they should be as close to each other as possible to get the greatest product. Let’s try x = 20 and y = 19.
Then xy = 20 × 19 = 380.
Problem 8. The rectangular region is 10 units by 8 units, resulting in a 9-by-7 array of lattice points in the interior of this
region. That’s 63 points with integer coordinates, as shown in the gure.
Problem 9. The ball rst drops 16 feet. It then travels up 8 feet and down 8 feet. When it hits the oor for the sixth time, it will have traveled
16 + 8 + 8 + 4 + 4 + 2 + 2 + 1 + 1 + 1/2 + 1/2 = 47 feet.
Problem 10. Any convex pentagon may be subdivided into three triangles, each with a total angle sum of 180 degrees. Thus, the sum of the interior
angles of any convex pentagon is 3 × 180 = 540 degrees. If the pentagon is regular, then each of its ve angles will have the same measure of 540 ÷ 5
= 108 degrees.
Warm-Up 4
Problem 1. If Danny’s opponent received 3 “parts” of the vote and Danny received 5 “parts” of the vote, then that’s 8 parts in all. Each part must be
312 ÷ 3 = 104, so there must have been 8 × 104 = 832 students who voted in the election.
Problem 2. The ratio of oranges to apples and bananas is 1:(2 + 3) = 1:5. Thus, if Charlie has 15 apples and bananas, he has 1/5 = x/15 ⇒ x = 3
oranges.
Problem 3. The sum of the integers from –30 to 30 is zero, so we need to nd only the sum of the integers from 31 to 50. Adding 31 + 50, 32 + 49,
etc., we get 10 sums of 81, which is a total of 810.
Problem 4. Six out of the 20 students, or 30%, scored less than 75%.
Problem 5. If we start with 35 pennies and systematically swap pennies for nickels, nickels for dimes, etc., we will î‚¿nd the following
24 combinations: 35P, 30P + 1N, 25P + 2N, 25P + 1D, 20P + 3N, 20P + 1N + 1D, 15P + 4N, 15P + 2N + 1D, 15P + 2D, 10P + 5N, 10P + 3N + 1D,
10P + 1N + 2D, 10P + 1Q, 5P + 6N, 5P + 4N + 1D, 5P + 2N + 2D, 5P + 3D, 5P + 1N + 1Q, 7N, 5N + 1D, 3N + 2D, 1N + 3D, 2N + 1Q and 1D + 1Q.
Problem 6. Each base in the expression is itself a power of 3, so we can simplify as follows: 3
1
× 9
2
× 27
3
× 81
4
⇒ 3
1
× (3
2
)
2
× (3
3
)
3
× (3
4
)
4
⇒ 3
1
×
3
4
× 3
9
× 3
16
⇒ 3
(1 + 4 + 9 + 16)
= 3
30
. The value of m is 30.
Problem 7. The idea is that the ratio of the 121 trout that were tagged and released on the î‚¿rst day compared with the unknown number of trout in the
lake is proportional to the 22 tagged out of 48 caught on the second day. We set up the proportion 121/x = 22/48, and solve for x. The cross product is
22x = 121 × 48, so x = (121 × 48)/22 = 264. We would estimate 264 trout in the lake.
Problem 8. The mixed number 3
1
/
3
is equal to the improper fraction 10/3. If we multiply 10/3 by its reciprocal 3/10, we get 1, which we could then
multiply by 4 to get 4. To do this in one step, we need the number 3/10 × 4 = 12/10 = 6/5 = 1
1
/
5
.
Problem 9. Since the area of square ABCD is 36 square inches, we know that each side is √36 = 6 inches, and the perimeter is 4 × 6 = 24 inches. We
are told that the perimeters of the square and rectangle are equal so the sides of the rectangle must add to equal half the perimeter, or 24/2 = 12 inches,
and must multiply to equal 20 square inches. Thus, the rectangle must be 10 inches by 2 inches, and 10 inches is the longest side.
Problem 10. Goats have 4 legs, and ducks have 2 legs. If twice the number of heads were equal to the number of legs, then it would be all ducks. The
extra 76 legs make 76 ÷ 2 = 38 pairs of legs that will turn 38 of our assumed ducks into goats.
Workout 2
Problem 1. The mean of a set of consecutive integers is the middle number, or in this case, halfway between the two middle numbers. We need to
look at sets of consecutive integers that are centered around 3.5. We also might recall that 6! = 720. After some experimentation, we î‚¿nd that the
consecutive integers are 1, 2, 3, 4, 5 and 6. Their product is indeed 720, and their sum is 21.
Problem 2. Mr. Adler will get a $5000 raise, and Mr. Bosch will get a $7000 raise so that they both have salaries of $45,000. Mr. Bosch’s salary was
raised by 7000 ÷ 38,000 × 100 = 18.4%.
Problem 3. We can see that Tim must have gotten at least 7 questions correct because 6 questions correct would have earned him a maximum of
only 5 × 6 = 30 points. If he got 7 questions correct, he earned 5 × 7 = 35 points so he could have gotten 2 questions wrong, for 2 × (–2) = –4, and 1
question blank to get him to 31 points. Thus, the ordered triple is (7, 2, 1).
Problem 4. The square root of 72,361 is 269. If we divide this by 3, we will be in the ballpark of our three consecutive primes. The primes are 83, 89
and 97, so the largest is 97.
Problem 5. The diameter of the sphere also must be 10 meters, so the radius is 5 meters. The formula for the volume of a sphere is (4/3)Ï€r
3
so the
volume of our sphere is (4/3) × π × 5
3
= (4/3) × π × 125 = (500/3)π. At this point, we have to decide what value of π to use. If we use π ≈ 3.14, we get
(500/3) × 3.14 ≈ 523.3 cubic meters to the nearest tenth. If we use π ≈ 3.142, we get (500/3) × 3.142 ≈ 523.7 cubic meters to the nearest tenth. If we
use π ≈ 3.1416, we get (500/3) × 3.1416 ≈ 523.6 cubic meters to the nearest tenth. After that, any additional digits of π will not change the result so we
should go with 523.6 cubic meters. Note: Rather than using approximations, the π key of your calculator should be used.
MATHCOUNTS 2008-200956
Problem 6. Using distance/rate = time, Anna runs to her friend’s house in (1mile/8mph) × 60 = 7.5 minutes. Anna runs back from her friend’s house
in (1mile/6mph) × 60 = 10 minutes. Thus, it takes her 10 − 7.5 = 2.5 minutes longer to run back from her friend’s house than it took her to run to her
friend’s house.
Problem 7. If the other leg and hypotenuse are consecutive whole numbers, let’s call them n and n + 1, respectively. Then we solve the Pythagorean
equation for n as follows: 29
2
+ n
2
= (n + 1)
2
⇒ 841 + n
2
= n
2
+ 2n + 1 ⇒ 841 = 2n + 1 ⇒ 840 = 2n ⇒ n = 420. This means the other leg is 420 units,
and the hypotenuse is 421 units. Their sum is 841 units, which is the square of 29. Incidentally, you can generate Pythagorean Triples in this way with
any odd number. Take the odd number as one leg, and split the square of the odd number into two consecutive whole numbers to get the other leg and
the hypotenuse.
Problem 8. There are 4
4
= 4 × 4 × 4 × 4 = 256 possible four-digit numbers that use the digits 2, 4, 6, 8 with repetition allowed. Of these, only 4! =
4 × 3 × 2 × 1 = 24 do not have a repeated digit. That means the other 256 – 24 = 232 numbers must have at least one of their digits repeated two, three
or four times. The probability that one of these is selected at random is 232/256 = 29/32.
Problem 9. Remy’s score of 204 is 46 points above his new average of 158. Since his new average of 158 is 2 points above his previous average of
156, we can imagine that the extra 46 points are distributed, 2 points each, among his previous 23 games. Now we know he has played 24 games. If
he wants his 25th game to bring his average up to 159, he will need a total of 25 × 159 = 3975 points. Right now he has 24 × 158 = 3792 points. The
difference, 3975 – 3792 = 183 points, is what he must bowl in the next game. We also could see that he needs to bowl the desired average of 159 plus
an extra 24 points to distribute among his previous 24 games, which is 159 + 24 = 183.
Problem 10. The positive square root of 200 is 10√2, or about 14.142. The positive square root of 121 is 11. Dividing 14.142 by 11, we get about
1.2856. Subtracting the 1 and multiplying by 100, we see that √200 is about 29% larger than √121.
Warm-Up 5
Problem 1. A distance of 1 meter is 50 times 2 cm, so the actual distance between the two cities is 50 times 2.5 km, which is 125 km.
Problem 2. If the ratio of the two complementary angles is 4 to 5, then there are 9 equal parts making up the full 90 degrees. That means each part is
10 degrees, and the two angles are 40 degrees and 50 degrees. When the 40-degree angle is increased by 10%, we get 44 degrees. The 50-degree angle
must drop down to 46 so that the two angles remain complementary. Dividing 46 by 50, we get 0.92, or 92%. The larger angle must decrease by 8%.
Problem 3. If we record the differences between consecutive numbers in the pattern, we get 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, etc. We could extend this pattern
to the 25th term, or we could try to think of a direct way to calculate the 25th term. We might realize that the 25th term is the second of two terms that
are 12 more than the previous term. If we start with 1, we need to add two times the sum of the numbers 1 through 12. This is 1 + 2(1 + 2 + ... + 11 +
12) = 1 + 2(6 × 13) = 1 + 12 × 13 = 157. (A sum of consecutive counting numbers starting with 1 is a triangular number. When we have two triangular
numbers, we can make “oblong numbers,” which are rectangles that are n by n + 1. Our 25th term is one more than the 12th oblong number, which is
1 + 12 × 13 = 157.)
Problem 4. If six numbers have a mean of 83.5, then the sum of the numbers is 6 × 83.5, which is 501. The ve known numbers have a sum of 419,
so the value of x must be 501 – 419 = 82. To nd the median of our six numbers, we arrange them in order from least to greatest as follows: 80, 82,
83, 84, 85, 87. The median is the average of 83 and 84, which is, coincidentally, 83.5.
Problem 5. To deal with the possibility of a second scoop, let’s just say that there are four avors for the second scoop, one of which is no extra
scoop. Thus, there are 2 cone options, 3 rst-scoop options and 4 second-scoop options which is 2 × 3 × 4 = 24 possible orders.
Problem 6. Let’s call the rst integer n. Then the second integer is 2n, and the third integer is 2 × 2n = 4n. The difference between 4n and n is 3n, but
17 is not a multiple of 3. The difference between 4n and 2n is 2n, but 17 is not a multiple of 2. The only difference that works is between 2n and n,
which is n, meaning that n is 17. The three numbers are 17, 34 and 68. Their sum is 119.
Problem 7. If the width of the rectangular playground is w, then the length is 2w + 25. A perimeter of 650 feet means the semi-perimeter is 325 feet.
The width plus the length equals the semi-perimeter, so w + 2w + 25 = 325. That means 3w = 300, so w = 100 feet. The length must be 2 × 100 + 25 =
225. The area of the playground is 100 × 225 = 22,500 square feet.
Problem 8. We need to look at equivalent ratios for both classes until we î‚¿nd two that have the same total boys as total girls. If the î‚¿rst class had 12
boys and 10 girls, and the second class had 3 boys and 5 girls, then we would have a combined total of 15 boys and 15 girls, which is a 1 to 1 ratio.
Unfortunately, this is not the correct solution because the second class would have fewer than 10 students. If we double the numbers of boys and
girls in this second class, we get 6 boys and 10 girls. We now need to go to the fourth multiple of the 6 to 5 ratio in the other class to make up for
the four fewer boys. Thus, the î‚¿rst class must have 24 boys and 20 girls. The number of students in the two classes is 30 boys and 30 girls, which
is 60 students. We also can solve this without using Guess and Check. The î‚¿rst ratio tells us there will be 6x boys and 5x girls for some value of x.
The second class will have 3y boys and 5y girls for some value of y. Together they will have 6x + 3y boys and 5x + 5y girls. When we set these equal
(getting our desired 1:1 ratio), we have 6x + 3y = 5x + 5y, which simpliî‚¿es to x = 2y. So if y = 1 then x = 2, and we have 12 boys/10 girls and
3 boys/5 girls. Unfortunately, there are not 10 students in the second class. So let’s do y = 2 and then x = 4, giving us 24 boys/20 girls and
6 boys/10 girls for a total of 30 boys and 30 girls, which is 60 students.
Problem 9. Suppose we put the two circles in opposite corners of the rectangle so that the circles are tangent to the
sides of the rectangle, and they are diagonally across from each other. Then the center of each circle is 3 inches in from
each side of the rectangle that it touches. Now imagine a rectangle that has opposite corners at the centers of these
circles. This smaller rectangle measures 8 inches by 6 inches. The diagonal of this rectangle is the greatest possible
distance between the centers of the two circles. It helps if we recognize that these lengths are 3 × 2 and 4 × 2, which
means we have a multiple of the 3-4-5 Pythagorean Triple. Thus, the length of the diagonal must be 5 × 2 = 10 inches.
Indeed, 8
2
+ 6
2
= 64 + 36 = 100 = 10
2
.
MATHCOUNTS 2008-2009 57
Problem 10. The prime factorization of 78 is 2 × 3 × 13, so the largest prime factor is 13.
Warm-Up 6
Problem 1. A square on the diagonal of another square has twice the area. Square II is 2 square units, and
square III is 4 square units.
Problem 2. Based on the 4-inch difference in their heights, Sue and Joe could be either 60 and 64 inches, respectively, or 64 and 68 inches. But the
girls are the three shortest students, so Sue has to be 60 inches and Joe is 64 inches tall. Since Steve is 1 inch shorter than John, he must be 68 inches
and John must be 69 inches. Mary is the shortest at 58 inches, so Lisa must be 63 inches tall. The sum of John’s and Lisa’s height is 69 + 63 =
132 inches.
Problem 3. The average of x, y and z is (x + y + z)/3. Twice the average of y and z is just the sum of y and z. Hence, we can write the equation
(x + y + z)/3 = y + z. Multiplying both sides of the equation by 3, we get x + y + z = 3y + 3z. Now we subtract y and z from both sides and î‚¿nd that
x = 2y + 2z, or x = 2(y + z).
Problem 4. Rearranging the equation to isolate y gives us y = 20x
2
+ 24. We could determine what y is for different possible values of x, but we
know that 20x
2
must be as small as possible. Since 20x
2
is a square, we know that it’s never negative. Thus, x must have the smallest absolute value
possible, which in this case, means that x = 0. Plugging 0 in for x, we î‚¿nd that the smallest value for y is 24.
Problem 5. If the current price of $30 is after a 50% reduction, then the Lux basketball shoes must have been 30 × 2 = $60 before. That $60 price
was after a 25% reduction. When 25% of the price is removed, the remaining portion is 75%, or 3/4 of the price, which means the original price
must have been $60 × 4/3 = $80 or $80.00.
Problem 6. The surface of the 2-by-4-by-8 rectangular solid is 2 × (2 × 4 + 2 × 8 + 4 × 8) = 2 × (8 + 16 + 32) = 2 × 56 = 112 square units. The
surface area of the 4-by-4-by-4 cube is 6 × 4 × 4 = 96. The difference, 112 – 96 = 16, is the number of painted unit-cube faces that are hidden in the
interior of the cube. To conrm this, let’s imagine how the unit cubes would be arranged. The 8 corners from the rectangular solid must become the
eight corners of the cube since these cubes each have three faces painted. The 32 cubes that were along the edges of the rectangular solid but not on
the corners have two faces painted. The 4-by-4-by-4 cube will need only 24 of these, which leaves 8 extra unit cubes with two faces painted. There
are 24 unit cubes from the centers of the largest rectangular solid’s faces that have only one side painted. The 4-by-4-by-4 cube will need exactly
these 4 for the centers of each face, which is a total of 24. That leaves the 8 extra unit cubes with two faces painted—a total of 8 × 2 = 16 unit-cube
faces—hidden in the very center of the 4-by-4-by-4 cube.
Problem 7. The 2009 calendar will start on a Thursday. Since 365 ÷ 7 leaves a remainder of 1, the 2010 calendar will start on a Friday. Similarly,
2011 will start on a Saturday, and 2012 will start on a Sunday. But there will be 366 days in 2012, which means that 2013 will start on a Tuesday.
Then 2014 will start on a Wednesday, and î‚¿nally 2015 start on a Thursday and will be the same calendar as 2009.
Problem 8. Two-thirds of 6 is 4, so there must be at least 4 questions for which true is the correct answer. If there are four Ts, then there are two Fs.
There are 6 × 5 ÷ 2 = 15 different patterns. If there are ve Ts, then there is just one F, and there are 6 different patterns (or places to put the F). If
there are six Ts, then there is just 1 pattern. That’s 15 + 6 + 1 = 22 different True/False answer patterns.
Problem 9. The value of (2 Ñ„ 4) is (1/2) + (1/4) + (1/2
2
) + (1/4
2
) = (1/2) + (1/4) + (1/4) + (1/16) = 17/16.
Problem 10. If the clever shopper takes $4 off followed by 20% off, the book will cost 0.8 × ($25 – $4) = 0.8 × $21 = $16.80. If she takes 20% off
followed by $4 off, it will cost (0.8 × $25) – $4 = $20 – $4 = $16.00. She will save $16.80 – 16.00 = $0.80 = 80 cents by taking the better-valued
approach.
Workout 3
Problem 1. If the car averages 60 mph in the rst 240 miles, then it took 240 ÷ 60 = 4 hours to do it. That leaves 616 – 240 = 376 miles and 9 – 4 =
5 hours to go. If the car averages 80 mph in the next 160 miles, then it took 160 ÷ 80 = 2 hours to do it. Now the car has 376 – 160 = 216 miles and
5 – 2 = 3 hours to go. The car must average 216 ÷ 3 = 72 miles per hour for the remainder of the trip.
Problem 2. Since 30/14 = 15/7 = 2
1
/
7
, the front wheels of the buggy must rotate more than twice as many times as the rear wheels. The rear wheels
cover 2 × 30 × π = 60π ≈ 60 × 3.14159 = 188.4954 inches of ground with each rotation. That’s about 63,360 ÷ 188.495 ≈ 336.13552 revolutions
in one mile or 188.4954 ÷ 12 = 15.70795 feet per revolution. The front wheels will cover 2 × 14 × � = 28� ≈ 28 × 3.14159 ≈ 87.96452 inches of
ground with each rotation. That’s about 63,360 ÷ 87.96452 ≈ 720.2904 revs in one mile. The positive difference in the number of revolutions of
the front and rear wheels is 720.2904 – 336.13552 = 384.15488 ≈ 384, to the nearest whole number. Ideally, students will solve the problem by
simplifying the following expression with their calculators and using the � button: (63,360 ÷ (28�)) − (63,360 ÷ (60�)) ≈ 384.1546, or 384 to the
nearest whole number.
Problem 3. At the moment that Rex completes the one-mile run, Stan has one minute to go and 1/6 of the distance, which is 1/6 × 5280 = 880 feet.
At that same moment, Tim has two more minutes to go and 2/7 of the distance, which is about 2/7 × 5280 = 1508.6 feet. Thus, Tim trails Stan by
1508.6 – 880 = 628.6 feet.
Problem 4. First of all, we should convert 480 inches to feet, which is 480 ÷ 12 = 40 feet. The rectangular solid box contains 2.75 × 4.05 × 40 =
445.5 cubic feet. One cubic yard measures 3 feet by 3 feet by 3 feet and contains 27 cubic feet. Thus, the box must contain 445.5 ÷ 27 =
16.5 cubic yards.
I
II
III
MATHCOUNTS 2008-200958
Problem 5.
Since the measure of angle BAC is 42, the other two equal angles of triangle ABC must be (180 – 42)/2 =
138/2 = 69 degrees each. We will introduce the point F, which is the center of the circle, and draw segments from each vertex
to F. Since segment ED is tangent to the circle at point C, it must be perpendicular to radius CF. Angle BAC is bisected by
segment AF, so angle FAC is 21 degrees. Angle FCA is also 21 degrees since triangle AFC is isosceles. Thus, the measure of
angle ACD is 90 – 21 = 69 degrees, which is the same as the two base angles of triangle ABC.
Problem 6. To î‚¿nd the sum of the positive odds less than 100, we imagine adding 1 and 9, 3 and 97, 5 and 95, etc. In all, we can make 25 pairs that sum
to 100, which is 2500. The positive, two-digit multiples of 10 contribute another 450, and the sum of 4, 16 and 64 is 84. In all, we have 2500 +
450 + 84 = 3034. There are 50 + 9 + 3 = 62 numbers in our data set, so the mean (average) is 3034 ÷ 62 = 48.935 to the nearest thousandth. The median
of the odds less than 100 is 50. When the two-digit multiples of 10 are added, the median is still 50. However, when 4 and 16 are added on the lesser side
and 64 is added on the greater side, the median of the 62 numbers becomes the average of 50 and 49, which is 49.5. The positive difference between the
median and the mean of the set of numbers is thus 49.5 – 48.935 = 0.565.
Problem 7. Ryosuke traveled a distance of 74,592 – 74,568 = 24 miles between the time he picked up his friend and when he dropped him off. Since his
car gets 28 miles per gallon, he used 24/28 or 12/14 of a gallon. At $4.05 per gallon, the cost of the trip is about 12/14 × 4.05 ≈ $3.47.
Problem 8. The prime factorization of 84 is 2
2
× 3 × 7, the prime factorization of 112 is 2
4
× 7, and the prime factorization of 210 is 2 × 3 × 5 × 7. The
greatest common factor of the three numbers is the product of all the prime factors that they have in common, which is 2 × 7 = 14.
Problem 9. If 10 men take 6 days to lay 1000 bricks, then 20 men should take 3 days to lay 1000 bricks. If 20 men are to lay 5000 bricks, then it should
take 5 × 3 = 15 days.
Problem 10. Ayushi must have 1 quarter and 5 pennies. If two coins are selected at random, there is a 5/6 × 1/5 = 1/6 chance that Ayushi will select a
penny and then the quarter. There is a 1/6 × 5/5 = 1/6 chance that Ayushi will select the quarter and then a penny. There is a 5/6 × 4/5 = 2/3 chance that
Ayushi will select two pennies. Only the last of these options amounts to less than 15 cents, so the probability is 2/3.
Transformation & Coordinate Geometry Stretch
Problem 1. Since only the y portions of the coordinates move, we know that the line of reection must be a horizontal line. Now we just need to nd the
midpoint between an original point and its reected image to pinpoint the location of the line. The y-coordinate of point A is 3 and the y-coordinate of A’
is −5; therefore, the midpoint is at (2, −1). The line of reection is y = −1.
Problem 2. We know that, for a triangle, area = 1/2(base)(height), which equals 30 in this problem. We also know that the height of the triangle is 4 if
we use the horizontal leg on the x-axis as the base. Now we can plug this information into the equation to î‚¿nd the length of the base that runs along the
x-axis. The equation is (1/2)(b)(4) = 30, so b = 30/2 = 15. Since the 3rd vertex is on the x-axis we know that it extends straight left 15 units from the
vertex at (0, 0), bringing us to the point (−15, 0).
Problem 3. When an image is translated to the right we just add the number of units it is being translated to the original x-coordinate. When an image
is translated down we just subtract that number of units from the y-coordinate. In this case we’ll subtract 2 from the y-coordinates and add 3 to the
x-coordinates. This will make point B(6, 5) move to B’(6 + 3, 5 − 2) = (9, 3).
Problem 4. In a regular pentagon each vertex is 360º/5 = 72º away from the adjacent vertices. This means that if a point is rotated counterclockwise
144º, it rotates 144/72 = 2 vertices counterclockwise. Thus, vertex C would land where vertex N is.
Problem 5. When an image is reected over an axis, the opposite coordinate changes sign. So if you translate over the y-axis, the x-coordinate changes
sign; and if you translate over the x-axis, the y-coordinate changes sign. In this case, we reect over both the y-axis and x-axis, causing both signs to
change. Point A was originally (−10, 2), which means the nal image has A at (10, −2).
Problem 6. When we rotate images 90º the coordinates switch places, and the signs are adjusted based on whether or not an axis was crossed. In this
case, rotating point A 90º will bring it across the y-axis into Quadrant I, which means both the x and y will be positive. The original point A was at (−4, 1)
so the nal image will be at (1, 4). We also could solve this problem by seeing that the slope of the segment from the origin to A is −1/4. If A is moving
to a location that is a 90º rotation about the origin, it will move to a point on the segment perpendicular to the one that currently connects it to the origin.
This will be the segment that has a slope of 4/1 or −4/−1 from the origin which puts us at (1, 4) or (−1, −4). The point (1, 4) is in the counterclockwise
direction we need.
Problem 7. By looking at the diagram provided, we can see that the line containing the point of rotation lands on top of itself, but the arrow is facing the
opposite direction. This tells us that 1/2 of a full 360º rotation was completed; therefore, the image rotated 360º/2 = 180º about point C.
Problem 8. Since the image is reected across the y-axis rst, we will just change the sign of the x-coordinate, which will give us (2, 6). Next the image
is shifted down 8 units so we will subtract 8 from the y-coordinate, giving our image a nal center of (2, −2).
Problem 9. With the center of dilation at the origin and a scale factor of 2, all the coordinates of square ABCD are twice the coordinates of its preimage.
The preimage has an area of 4 square units, so its side length is 2 units. Since the center of the preimage is at (8, –8), the four vertices of the preimage
are at (7, –9), (7, –7), (9, –7) and (9, –9). The point (9, –9) is the farthest from the origin on the preimage, so the point farthest from the origin on the
image of square ABCD is (18, –18).
Problem 10. Lines that are parallel have the same slope. In this case, AB has a slope of (0 − (−4))/(−4 − 0) = −1. This now must be the slope for XY.
Now we can use the equation y
2
− y
1
= m(x
2
− x
1
) to nd the value of k. Plugging in the coordinates for Y and X we nd that k − 8 = −1(14 − 0), thus k =
−14 + 8 = −6. We also could see that from (0, 8) to (14, k) we are moving 14 units right, so we also must move 14 units down to get a slope of −14/14 =
−1. Moving 14 units down from (0, 8) lands us at (0, 8 − 14) or (0, −6), so k = −6.
A
B
C
D
E
F
MATHCOUNTS 2008-2009 59
Problem Index
Algebraic
Expressions
& Equations
WU 2-6
WU 2-9
WO 1-4
WO 1-6
WO 1-10
WU 3-7
WU 4-3
WU 4-6
WO 2-3
WO 2-4
WU 5-6
WU 6-3
WU 6-4
WU 6-9
WU 7-2
WU 8-7
WU 8-10
WU 9-2
WU 9-4
WU 9-9
WU 10-2
WU 10-10
WU 11-3
WU 11-7
WU 12-8
WO 6-1
WU 13-6
WU 13-9
WU 14-1
WU 14-4
WU 15-9
WU 16-4
WU 16-7
WO 8-5
WO 8-9
WO 9-2
Sequences &
Series
WU 3-2
WO 4-10
WU 10-9
WO 6-4
WU 13-3
WU 15-5
WU 15-10
WU 17-8
WU 18-3
WO 9-6
Measurement
WU 1-2
WU 1-5
WU 2-2
WU 2-7
WU 2-10
WO 1-2
WO 1-8
WU 3-1
WO 2-7
WU 5-7
WO 3-2
WU 7-10
WO 4-1
WO 4-6
WU 9-3
WU 9-10
WU 10-4
WO 5-9
WU 11-9
WU 12-2
WO 6-2
WU 14-5
WU 16-3
WU 16-9
WO 8-3
WU 18-2
Number
Theory
WU 1-4
WU 1-7
WU 1-8
WU 2-3
WO 1-7
WO 2-8
WU 5-10
WO 3-8
WU 7-3
WU 7-6
WU 8-3
WU 8-5
WO 4-5
WU 9-1
WU 10-6
WU 10-7
WO 5-4
WO 5-6
WU 13-2
WU 13-7
WU 14-2
WU 14-7
WU 15-7
WO 8-6
WU 17-3
WU 17-10
Solid Geometry
WO 2-5
WU 6-6
WO 3-4
WU 8-8
WO 5-2
WO 6-7
WO 8-8
WO 9-7
Plane
Geometry
WO 1-5
WU 3-10
WU 4-9
WU 5-9
WU 6-1
WO 3-5
WU 7-7
WO 4-3
WO 4-9
WU 9-8
WU 12-4
WO 6-3
WU 13-4
WU 13-8
WO 7-2
WO 7-7
WU 15-3
WU 15-6
WU 16-6
WU 16-8
WO 8-7
WO 8-10
WU 17-2
WU 17-6
WU 18-6
WO 9-4
Coordinate
Geometry
WO 1-9
WU 3-8
*Transformations &
Coord. Geom. Stretch
WU 7-5
WU 8-6
WO 4-2
WO 4-8
WO 5-5
WU 11-1
WU 11-10
WO 6-5
WO 7-6
WO 7-10
WU 15-1
WU 18-10
WO 9-1
Logic
WU 1-9
WU 2-5
WU 4-10
WU 6-2
WU 6-7
WU 12-10
WU 13-1
WU 14-10
WO 7-3
WU 15-2
WU 16-10
WO 8-4
WU 18-1
WO 9-5
Probability,
Counting &
Combinatorics
WU 1-3
WU 2-4
WU 3-6
WO 3-10
WU 7-9
WU 8-4
WU 10-1
WU 13-5
WU 14-3
WU 17-5
WU 17-9
WU 18-5
WO 9-10
Percents/Fraction
WO 1-3
WU 3-3
WU 4-8
WO 2-10
WU 5-2
WU 6-5
WU 6-8
WO 4-4
WU 9-7
WU 10-8
WO 5-3
WO 5-8
WU 11-8
WU 12-3
WU 12-6
WO 6-6
WU 13-10
WU 14-8
WO 7-4
WO 8-1
WU 18-7
WO 9-9
Pattern
Recognition
WU 2-1
WU 5-3
WU 12-5
WU 18-9
WO 9-3
Proportional
Reasoning
WU 1-1
WU 3-4
WU 3-5
WU 4-1
WU 4-2
WU 5-1
WO 3-9
WU 7-1
WU 8-1
WU 9-6
WU 10-5
WU 11-4
WU 11-5
WU 12-7
WO 6-9
WU 14-6
WU 14-9
WO 7-1
WU 15-4
WO 8-2
WU 18-4
WU 18-8
It is difî‚¿cult to categorize many of the problems in the MATHCOUNTS School Handbook. It is very
common for a MATHCOUNTS problem to straddle multiple categories and hit on multiple concepts. This
list is intended to be a helpful resource, but it is in no way complete. Each problem has been placed in
exactly one category.
**Please note that the problems in regular type can be found in Volume I of the handbook, while the
problems in italics can be found in Volume II of the handbook.
Problem Solving
(Misc.)
WU 1-10
WU 3-9
WU 4-7
WO 2-6
WU 5-5
WU 5-8
WU 6-10
WO 3-1
WO 3-3
WU 7-4
WU 8-9
WO 5-7
WU 11-2
WU 11-6
WU 12-9
WO 6-10
WO 7-8
WU 15-8
WU 16-2
WU 17-4
WO 9-8
General Math
WU 1-6
WU 2-8
WO 1-1
WU 4-5
WO 2-2
WO 3-7
WU 9-5
WO 5-1
WO 5-10
WU 12-1
WO 7-5
WO 7-9
WU 16-1
WU 16-5
WU 17-1
WU 17-7
Statistics
WU 4-4
WO 2-1
WO 2-9
WU 5-4
WO 3-6
WU 7-8
WU 8-2
WO 4-7
WU 10-3
WO 6-8
*The Mixture Stretch and Triangles ULTRA Stretch in Volume II have 20 more problems that also include geometry,
algebra, measurement and proportional reasoning concepts.
60 MATHCOUNTS 2008-2009
noTeS
MATHCOUNTS 2008-2009 61
noTeS
62 MATHCOUNTS 2008-2009
noTeS
MATHCOUNTS 2008-2009 63
MATHCOUNTS CLUB PROGRAM
This year, MATHCOUNTS is building on the success of the MATHCOUNTS Club Program (MCP),
which was introduced last year as part of our 25th anniversary celebration. The MCP may be used by
schools as a stand-alone program or incorporated into preparation for the MATHCOUNTS Competition
Program. The MCP provides schools with the structure and activities to hold regular meetings of a math
club. Depending on the level of student and teacher involvement, a school may receive a recognition
plaque or banner and may be entered into a drawing for prizes. Open to schools with 6th-, 7th- and 8th-
grade students, the Club Program is free to all participants.
This year, the grand prize in the Gold Level drawing is an all-expenses paid trip for the club coach and
four students to Orlando, Florida to watch the 2009 Raytheon MATHCOUNTS National Competition
held at the Walt Disney World Swan and Dolphin Resort. All schools who have successfully completed
the Ultimate Math Challenge will attain Gold Level status and will be eligible for the grand prize drawing.
MATHCOUNTS COMPETITION PROGRAM
MATHCOUNTS proudly presents the 26th consecutive year of the MATHCOUNTS Competition
Program, consisting of a series of School, Local (Chapter), State and National Competitions. More
than 6,000 schools from 57 U.S. states and territories will participate in this unique mathematical bee.
The î‚¿nal 228 Mathletes will travel to Orlando, Florida, May 7-10, to compete for the prestigious title
of National Champion at the 2009 Raytheon MATHCOUNTS National Competition. Schools may
register via www.mathcounts.org or on the following registration form.
Do not hold up the mailing of your Volume II because you are waiting for a purchase order to
be processed or a check to be cut by your school for the registration fee. Fill out your Request/
Registration Form and send in a photocopy of it without payment. We immediately will mail your
Club in a Box resource kit (which contains Volume II of the MATHCOUNTS School Handbook)
and credit your account once your payment is received.
SPECIAL COACHES' COMMUNITY ON MATHCOUNTS.ORG
Registered Competition Program and/or Club Program Coaches will receive special access to the
Coaches' Community on www.mathcounts.org. If you have not already done so, you will need to
create a new User Proî‚¿le and allow MATHCOUNTS to verify your User Proî‚¿le request. For more
information about Coaches' Community access, refer to page 8 of the Club Resource Guide or
page 5 of Volume 1 of the 2008-2009 MATHCOUNTS School Handbook (page 6 of Volume 2).
2008-2009 Club Program Material Request
and Competition Registration
Why do you need this form?
Sign Up a Math Club and receive the Club in a Box resource kit [• FREE]
Register Your School to Participate in the MATHCOUNTS Competition Program •
[Competition Registration Deadline is Dec. 12, 2008]
NEW!
64 MATHCOUNTS 2008-2009
Payment:  Check  Money order  Purchase order # _____________ (p.o. must be included)   Credit card
Name on card: _______________________________________________________________  Visa  MasterCard
Signature: _________________________________________ Card #: ____________________________________ Exp: __________
Make checks payable to the MATHCOUNTS Foundation. Payment must accompany this registration form. All registrations will be conî‚¿rmed with an invoice
indicating payment received or payment due. Invoices will be sent to the school address provided. If a purchase order is used, the invoice will be sent to the address on the purchase
order. Payment for purchase orders must include a copy of the invoice. Registration questions should be directed to the MATHCOUNTS Registration Ofî‚¿ce at 301-498-6141.
Registration conî‚¿rmation may be obtained at www.mathcounts.org.
REQUEST/REGISTRATION FORM: 2008-2009 School Year
Option 1
 SIGN UP MY MATH CLUB for the MATHCOUNTS Club Program and send me the
Club in a Box resource kit with Volume II of the 2008-2009 MATHCOUNTS School Handbook which
contains 200 math problems. (There is NO COST for the Club Program.)
(NOTE: You must complete the survey below.) Please see page 19 in Vol. 1 of the School Handbook or visit the Club
Program section of www.mathcounts.org for details.
Is this the rst year for a Math Club at your school?  Yes  No (If no, for how many years has there been a Math Club? ______)
# of Students in Math Club: _________ On average, how many MATHCOUNTS problems will you use each month? _________
How often do you expect the Math Club to meet during the fall/winter?
 Daily  2-4 Times Per Week  Weekly  2-3 Times Per Month  Monthly  Less Than Once a Month
When does the Math Club meet? (please select all that apply)  Before School  During School  After School  Weekends
What MATHCOUNTS resources do you use in your classroom with some/all of your classes? (please select all that apply)
 School Handbook  Problem of the Week  Prior Competitions  Prior School Handbooks  N/A
On average, how many MATHCOUNTS problems do you use in the classroom with some/all of your classes each month? _________
How many students work these problems during class? _________ Does your school have a MATHCOUNTS class?  Yes  No
How relevant are the MATHCOUNTS problems to your state curriculum expectations for grades 6, 7 and 8?
 Not at all; irrelevant topics  Not at all; too difcult  Somewhat  Very  Unsure
Mail or fax this completed form (with payment if choosing Option 2) to: MATHCOUNTS Registration
P.O. Box 441, Annapolis Junction, MD 20701
Fax: 301-206-9789
Teacher/Coach’s Name Principal’s Name ____________________________________
School Name  Previous MATHCOUNTS School
School Mailing Address
City, State ZIP County
School Phone ( ) School Fax # ( )
Teacher/Coach's Phone ( ) Chapter
(if known)
Teacher/Coach’s E-mail
What type of school is this?  Public  Charter  Religious  Private  Homeschool  DoDDS  State Dept.
Competition Registration (Option 2) must be postmarked by Dec. 12, 2008
Competition Registration Fees:
 Team Registration (up to four students)
 Individual Registration(s): # of students ____ (max. of 4)
By completing this registration form, you attest to the school administration’s
permission to register students for MATHCOUNTS under this school’s name.
Rate
1 @ $80 = $ ______
@ $20 each = $ ______
Total Due = $ ______
Title I Rate*
1 @ $40 = $ ______
@ $10 each = $ ______
Total Due = $ ______
 REGISTER MY SCHOOL for the MATHCOUNTS Competition Program
and send me the Club in a Box resource kit and Volume II of the 2008-2009
MATHCOUNTS School Handbook.
(NOTE: You must complete the survey below.) Please see
page 9 in Vol. 1 of the School Handbook or visit the Competition Program section of www.mathcounts.org.
Option 2
* Principal Signature is required to verify school qualiî‚¿es for Title I fees:
ALL must complete the survey portion of the Request/Registration Form