2015-2016
PLAYbook
2015–2016
Playbook
Contains 250 creative math problems
that meet the NCTM Grades 6-8 Standards.
2016 MATHCOUNTS
National Competition Sponsor
National Sponsors:
Raytheon Company
Northrop Grumman Foundation
U.S. Department of Defense
National Society of Professional Engineers
CNA Foundation
Phillips 66
Texas Instruments Incorporated
3Mgives
Art of Problem Solving
NextThought
Change the Equation has recognized MATHCOUNTS
as having one of the nation’s most effective STEM
learning programs, listing the Math Video Challenge as
an Accomplished Program in STEMworks.
Executive Sponsors:
General Motors Foundation
The National Council of Examiners for
Engineering and Surveying
Official Sponsors:
Bentley Systems Incorporated
Expii, Inc.
Associate Sponsors:
Edward T. Bedford Foundation
PricewaterhouseCoopers LLP
Tableau Software
Founding Sponsors:
National Society of Professional Engineers
National Council of Teachers of Mathematics
CNA Foundation
The National Association of Secondary School
Principals has placed all three MATHCOUNTS
programs on the NASSP Advisory List of National
Contests and Activities for 2015-2016.
HOW TO USE THIS
PLAYBOOK
If You’re a New Team Advisor
Welcome! We’re so glad you’re a team advisor this year.
Check out the Guide for New Team Advisors
starting on the next page.
If You’re a Returning Team Advisor
Welcome back! Thank you for leading a team again.
Click ahead to the 2015-16 Program Materials
starting on pg. 13.
Guide for New Team Advisors
Welcome to the Math Video Challenge! Thank you so much for serving as a team advisor this year. We’ve created
this Guide for New Team Advisors to help get you acquainted with the Math Video Challenge and to explain your role
in this program.
If you have questions at any point during the program year, please feel free to contact the MATHCOUNTS national
office at (703) 299-9006 or [email protected].
The Math Video Challenge in a Nutshell
The Math Video Challenge is a national program that challenges students to develop their math, communication
and technology skills in a collaborative video project. Created in 2011, it is a completely free program and is open to
all sixth-, seventh- and eighth-grade students. Students can participate through their school or through a non-school
group.
HOW DOES IT WORK? The Math Video Challenge is designed to be a fairly flexible program, meaning your students will spend more or less time participating in the program depending on how much they decide to do with their
video project. Here’s what a typical program year looks like.
During the fall/winter students work in teams of 4 to create a video based on a
MATHCOUNTS problem. The students must solve a problem from the 2015-2016 School
Handbook (pg. 15 of this playbook) and show a real-world application of the math concept
used in the problem. Videos can be no longer than 5 minutes; most videos are between 3 and 5
minutes in length. Students post their completed videos to the Math Video Challenge website.
From early February through mid-March, the general public votes on the videos and
the 100 videos with the most votes advance to the judging rounds. Students can vote
for their own video as often as once a day during general public voting, and they also can ask
their friends, family members and school community to vote.
20
A panel of expert judges reviews the top 100 videos and selects 20 semifinalists to
advance to the second round of judging. Semifinalists are announced in late March; semifinalists receive certificates and recognition on the program website.
In early April, expert judges select 4 finalist videos to advance to the Math Video
Challenge Finals. The 16 students who created these 4 videos, as well as each team’s advisor, receive an all-expenses-paid trip to the MATHCOUNTS National Competition, where they
present their videos to the 224 Competition Series national competitors.
At the Math Video Challenge Finals in May, the 224 Mathletes at the National
Competition vote to determine the First Place Video. The 4 students who created the
winning video each receive a $1,000 scholarship. All videos created during the contest are
added to the Math Video Challenge online archive, providing a free educational tool for
students and teachers.
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2015-2016 MATHCOUNTS Playbook
WHAT IS EXPECTED OF THESE STUDENT-MADE VIDEOS? Each video is different and your students will no
doubt come up with lots of different creative ideas for their video. The Math Video Challenge gives students a lot of
flexibility when it comes to the layout, production and content of their videos. The only requirements for the student
videos are:
Video Solves a
Video Is Created by a
Team of 4 U.S. Students Problem from the 20152016 MATHCOUNTS
in 6th, 7th and/or
School Handbook
8th Grade
More students can appear
in the video and the team
can seek help from others,
but only 4 students are
official team members for
a video.
Students can choose any
of the 250 problems from
this year’s handbook
(pg. 15). The video must
explain the solution to the
problem.
Video Shows a
Real-World
Application of Math
Video Is No More
Than 5 Minutes Long
and Contains No
Copyrighted Material
Videos must show how
Videos cannot include
the math concept explored copyrighted material—such
in the handbook problem as music, images or video
could play out in the real clips—and must meet time
world.
requirements.
When trying to gauge the calibre and general format of videos submitted to this contest, a good way to start is to
watch videos in the Math Video Challenge Archive, including past semifinalists, finalists and winners. This archive
is available at videochallenge.mathcounts.org/videos. You’ll notice that the quality and style vary significantly from video to video, but all submitted videos are clearly student-made. It’s a good idea to show your students some archived
videos, so the idea of creating a video does not seem daunting.
HOW ARE VIDEOS EVALUATED IN THE CONTEST? In the first round of the contest the top 100 videos are
determined solely based on general public voting. The 100 videos that receive the most votes between February 4
and March 14 (Pi Day) advance to the judging rounds. During the judging rounds, videos are evaluated based on the
following criteria, in order to select the 20 semifinalists and then the 4 finalists.
Creativity
Is the style well-suited
for the audience/middle
school students?
Is the story engaging?
Is the video memorable?
Does the video show
imagination?
Communication
Mathematical
Accuracy
Is the math communicated Is an appropriate approach
clearly and logically?
to solving the problem
Is time used effectively?
used?
Are important ideas
Are there any errors?
emphasized?
Is the solution explained
Is the video easy to
clearly?
understand and follow?
Real-World
Application
Does the video present a
clear, real-world application of the math concepts
in the problem?
Is the real-world
scenario believable?
A detailed sample assessment tool is included on pg. 47 of this playbook. You can use this to evaluate the video your
students create if you are using the Math Video Challenge as a group project assignment in your classroom instruction, for example, or give it to your students as a guide.
2015-2016 MATHCOUNTS Playbook
3
The Role of the Team Advisor
Your role as the team advisor is such an important one, but that doesn’t mean you need to be a math expert or treat
advising a team like a full-time job. Every team is different and you’ll find the style that works best for you and your students. Some students may need more help completing this project than others. But in general every MATHCOUNTS
team advisor must do the following.
• Initiate the registration process online and oversee the process to ensure parent/guardian approval is obtained
for all 4 student team members.
• Provide guidance and suggestions, as needed, to the students working on videos.
• Ensure students are aware of program deadlines (such as general public voting and video submission).
• Review the students’ video to ensure the content is appropriate and approve the video online so it can be officially
submitted to the contest.
• Accompany the students to the National Competition, if the students’ video is selected as a finalist in the contest.
You don’t need to know how to solve every MATHCOUNTS problem to be an effective team advisor. In fact, working
through the solution carefully is a crucial part of the Math Video Challenge program. Chances are, you’ll learn with and
alongside your students throughout the program year.
You don’t need to spend your own money to be an effective team advisor.
Your students do not need expensive video equipment to participate in this
program. Many videos submitted to the contest are filmed on smartphones.
There are also lots of free video editing software programs you and your students can use. Many students and teachers also have used equipment and
software for free through their school’s media center or a public library.
And, if you find that you want to do more with your students and need additional funding, you and your team can participate in Solve-A-Thon to raise
money for your MATHCOUNTS program. Learn more about this fundraiser at
solveathon.mathcounts.org.
The next sections of this Guide for New Team Advisors will go into more detail
on what participation in this program entails and how to create a video using
the free resources provided by MATHCOUNTS.
Check out Solve-A-Thon at
solveathon.mathcounts.org
to learn a free and easy
way to raise extra funds
for your video
team.
What Participation Means in This Contest
The Math Video Challenge is designed to work for lots of different groups. What participation looks like for your students will depend, to some extent, on how far they hope to advance in the contest. This section will go into a little more
detail about team formation and registration, as well as video submission and approval.
WHAT DO I NEED TO KNOW ABOUT FORMING A TEAM? Because of the nature of this contest, the way Math
Video Challenge teams are formed is different from typical math competitions. Here is what is most important to keep
in mind when establishing a team for this program.
• A team in the program consists of exactly 4 students.
• Students can be on more than 1 team. For example, if you have 6 interested students, they could form 2 teams,
with 2 students participating on both teams.
• A team can create more than 1 video. A team of students can make as many videos as they like.
• Teams cannot combine to make 1 video together. The minimum number of videos created per team is 1.
• A team advisor can oversee more than 1 team. For example, some math teachers will make the Math Video Challenge a group project that is included in their classroom instruction. In these cases, 1 teacher serves as the team
advisor for multiple teams.
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2015-2016 MATHCOUNTS Playbook
When forming a team, it’s a good idea to consider different students’ skills, interests and strengths. For example, one
student may take the lead on script writing, while another student may write an excellent solution to the problem. Part
of what makes this program great is that it allows students to collaborate and showcase a diversity of talents...take
advantage of this when forming teams!
HOW DO I REGISTER A TEAM? The online nature of this contest necessitates that each of the 4 student team
members get permission from a parent/guardian to participate. This adds an extra step to the registration process, but
it’s a really important one. Here’s how the Math Video Challenge registration process works:
Team
Advisor
Team advisor registers at videochallenge.mathcounts.org/register.
The team advisor must have a working email address in order to register a team of students for this program.
Team advisor creates a new team on the program website.
A team advisor can lead more than 1 team. If you are leading multiple teams, you would
repeat this step (and each subsequent step) for each team, so all of your teams are
linked to your team advisor account.
Team advisor requests parent/guardian permission for each student team
member—either online or offline.
When you create a new team, you’ll indicate how each team member’s parent will give
permission for his/her child to participate.
• Online Permission: The team advisor provides the parent’s email address online.
MATHCOUNTS emails the parent a link to an online permission form.
• Offline Permission: The team advisor provides the parent’s first and last name online
and then gives the parent the Offline Permission Form (the form is included in this
playbook on pg. 46).
Parent/guardian of each student team member completes the permission
form—either online or offline—and submits it to MATHCOUNTS.
Instructions for submitting this form are provided to the parent/guardian.
Team’s registration is complete once parent permission is received for all
students on the team.
MATHCOUNTS will email a notification to the team advisor once the team’s registration
is complete.
WHAT DO I NEED TO KNOW ABOUT SUBMITTING AND APPROVING A VIDEO? After your team’s registration is complete, you’ll receive an email confirming that your team is registered and can submit videos online for the
contest. Included in this email will be an access code for uploading videos, which is unique to your team (if you
are leading multiple teams, then you’ll get an emailed access code for each team you register).
Once your students have completed their video (refer to the next section of this guide for detailed information about
creating a video for the contest), either a student team member or you as the team advisor can upload it to the Math
Video Challenge website.
If A Team Advisor Submits the Video: The team advisor should be logged into the Math Video Challenge website.
If the team registration has been completed, then the Upload a Video button will appear. Click
the button to upload your video to the contest site. After the video upload is complete, you will
need to approve the video separately. You cannot approve and submit the video at the same
time, and the site will require that you watch the video in its entirety before approving it.
2015-2016 MATHCOUNTS Playbook
5
If A Student Team Member Submits the Video: In order for a student team member to upload the team’s video,
you must share the access code for uploading videos with him/her. The access code is included in an email sent to
you after your team’s registration is complete, and if you are logged in as a team advisor, it also is listed on your account next to the team’s information.
Once the student has the access code, s/he should go to
videochallenge.mathcounts.org/upload-video. S/he should
enter the access code and then click the Go to Upload Form
button. From there, the student will be able to upload the
video to the contest site.
Once the student has uploaded the team’s video, you will
receive an email notification that the video needs to be approved. To approve the video, log into your team advisor account. You will need to watch the video in its entirety before
you can approve it.
It’s important to check the content of the students’ video before approving it online. As the team advisor, you must
review the video to ensure that it is appropriate and follows the contest rules.
Producer’s Checklist for Creating a Video
Creating a video is so much fun and allows students to tap into their technology, art and communications talents
while building their math skills at the same time. Although your students will create the video for this program, you as
the team advisor will likely need to guide your students or give them suggestions—especially if your students hope to
advance to the semifinals or finals.
To make it easier to keep track of what video production entails, we have created a sort of Producer’s Checklist,
based on feedback from past participants about their video making process. We recommend reading through the
checklist yourself and sharing its contents with your students.
Please note, these steps are suggestions—not requirements. Different teams will have different methods for
creating a video and you’ll find what works best for you and your students. This checklist should be considered a
resource for the Math Video Challenge, not a list of requirements.
THE 10 MOST IMPORTANT STEPS WHEN CREATING A VIDEO:
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2015-2016 MATHCOUNTS Playbook
1. Pick a math problem.
Use this year’s handbook problems on pg. 15.
2. Write the solution(s) to the math problem
Explore different solutions to the problem.
6. Write the script for your video.
Keep the 5-minute time limit in mind.
7. Do a timed table-read and
run-through to practice.
Practice a few times before filming anything!
3. Brainstorm story ideas.
Use the graphic organizers on pgs. 49-50.
8. Shoot your video.
Be sure to avoid common pitfalls!
Some are listed on pg. 11.
4. Plan out the where, when and how of
filming, costumes, sets and editing.
Get the logisitics in order before getting
too far along in the project.
9. Edit your video and add
desired special effects.
Make sure your video does not include
any copyrighted material!
TO DO
5. Create a storyboard for your video.
Use the storyboarding template on pg. 51.
10. Show the final product to someone
who hasn’t seen the video before.
Based on his/her feedback, make any
necessary changes.
The remainder of this section (pgs. 8-12) provides a little more detail about each of these 10 steps.
2015-2016 MATHCOUNTS Playbook
7
STEP 1: PICK A MATH PROBLEM. This may sound like an obvious step, but choosing your problem carefully is
very important. When your students are deciding which math problem they want to solve in their video, it’s important
they think about questions like:
•How would we explain the solution to this problem in our video? How long would it take to
explain it well?
•Is there a real-world application we can imagine for this problem? Do we want to create our
own real-world scenario or use a problem that already has some sort of scenario?
•Can we come up with an interesting and memorable story for this math problem?
It’s important that your students choose only from the 250 problems in this year’s handbook. You can
find the problems in this playbook, starting on pg. 15. You’ll notice that the playbook problems do not go in sequential
order. This is because we have modified the 2015-2016 MATHCOUNTS School Handbook to work better for the
Math Video Challenge. Instead of separating problems into Warm-Ups, Workouts and Stretches—which are intended
for the MATHCOUNTS Competition Series—we have separated them into categories by math topic. If you and your
students would rather use the traditional handbook setup, you can access it at www.mathcounts.org/handbook.
STEP 2: WRITE THE SOLUTION(S) TO THE MATH PROBLEM. After your students
have decided which problem they want to solve, they should write a detailed, complete
solution, which will be the basis of their video. The answers to all 250 problems are included in this playbook starting on pg. 39.
Use the free Interactive
MATHCOUNTS Platform at
mathcounts.nextthought.com
to get detailed solutions
for this year’s handbook
problems!
If your students are having difficulty working through the solution to their
problem, a great resource is the Interactive MATHCOUNTS Platform,
powered by NextThought. This free online resource allows students to work
through current and past MATHCOUNTS problems, chat with other students about strategies and get step-by-step solutions to all 250 of this
year’s handbook problems. Students and team advisors can sign up for free
at mathcounts.nextthought.com.
Many MATHCOUNTS problems can be solved in different ways. Encourage
your students to think through multiple ways to explain the solution to the
problem, so they are sure to include the one they believe is best. However,
including multiple solutions in the video may make it more memorable, interesting or clear.
STEP 3: BRAINSTORM STORY IDEAS. Let your students’ creativity loose during this step! Have students think
of different scenarios or stories they could use to give context to the math problem they chose. We have included a
few graphic organizers you and your students can use to map out ideas (pgs. 49-50).
This step also lends itself to collaboration, as the students will need to work together to come
up with an idea everyone is excited about. Encouraging students to cooperate and listen to
each other’s ideas will help them come up with great ideas for their video, and also will set
them up for success when they need to collaborate later on the filming and editing of their
video.
It’s really important to stress to the students that they do not have to use the context provided in a
particular problem to create their video. Students can change the names of the characters, locations, props
or objects discussed, and even the activity the characters in the problem are doing. As long as the same problem is
solved, and it is clear which handbook problem the video explains, the students are allowed to change the context of
a handbook problem for their video.
On the next page are a few examples of how a problem could be changed for this program.
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2015-2016 MATHCOUNTS Playbook
Problem #62 from the 2014-2015 MATHCOUNTS School Handbook:
• Original Problem Text: The Statesville Middle School basketball team has 8 players. If a player can play any position, in how many different ways can 5 starting players be selected? (Answer: 56 ways)
• Example of Acceptable Modified Text: Molly has 8 different paint colors. If she can use any 5 colors for a painting she is creating, how many different ways can 5 colors be selected? (Answer: 56 ways)
• Example of Unacceptable Modified Text: The Statesville Middle School basketball team has 8 players. If 5 players are needed for a game, how many players are not selected in any given game? (Answer: 3 players)
You’ll notice in the acceptable example, the basics of the problem are kept intact. The difficulty level, math concept
and steps to solve are not affected by changing the middle school basketball team with 8 players to a girl with 8 colors of paint. In contrast, the change made in the unacceptable example has a significant impact on the problem—it is
easier and explores a different math concept from the original.
Problem #219 from the 2014-2015 MATHCOUNTS School Handbook:
• Original Problem Text: Ronny had 9 oranges, and Donny had 15 oranges. They met up with Lonny, who had no
oranges. Lonny gave $8 to Ronny and Donny, and the three of them shared the oranges equally. If Ronny and
Donny split the $8 in proportion to the number of oranges each contributed, how much of the $8 should Ronny
receive? (Answer: $1)
• Example of Acceptable Modified Text: Samantha had 9 pencils, and Kaleb had 15 pencils. They met up with
Ford, who had no pencils. Ford gave $8 to Samantha and Kaleb, and the three of them shared the pencils equally.
If Samantha and Kaleb split the $8 in proportion to the number of pencils each contributed, how much of the $8
should Samantha receive? (Answer: $1)
• Example of Unacceptable Modified Text: Ronny had 12 oranges, and Donny had 2 oranges. If the two of them
want to share the oranges equally and each orange costs $1.50, how much should Donny pay Ronny to reimburse Ronny for the oranges he gave to Donny? (Answer: $7.50)
Again, you’ll notice in the acceptable example, the basics of the problem are kept intact. The difficulty level, math
concept and steps to solve are not affected by changing the names and objects in the problem. In contrast, the
change made in the unacceptable example makes the modified problem so different from the original problem that it
is unrecognizable as Problem #219.
STEP 4: PLAN OUT THE WHERE, WHEN AND HOW OF FILMING, COSTUMES, SETS AND EDITING. It’s
important that your students plan out the logistics of their video project, and it can be easy for tasks to fall through the
cracks. After your team has finished the imaginative side of their project, have them plan out how they will actually do
this project. Here are some things to think about:
Costumes
• Do we want costumes or is wearing our regular clothing good for our story?
• If we will be using costumes, do we need to find or buy materials to make them?
• If we will be using costumes, do we have someone who knows how to sew (or glue)
them together?
TO DO
Sets and Location
• Do we need to create any sets or backgrounds for our video?
• Could we create backgrounds or sets in the video editing stage using moviemaking software?
• Is there a place we can go to film our video that is the right setting for our story, so we can minimize the number
of sets we need to create?
• Would it be easier for us to shoot the film inside or outside?
• Do we need to make arrangements with someone in order to film at a particular location?
Filming and Equipment
• Are we going to film the entire video in one day or space it out? Do we need to reserve space for extra days or
make arrangements for our costumes and sets if we’re filming for longer?
2015-2016 MATHCOUNTS Playbook
9
•
•
•
•
•
Are we going to film the scenes in order, so it’s easier to edit? Are we going to film the scenes as they are ready
so we can possibly finish filming sooner?
Are there days that work for everyone on the team?
Does every team member need to be present for all days of filming?
Do we have a camera (or a phone with a camera) we can use to shoot our video?
If we do not have a camera, can we borrow one from our school’s media center or the local public library? What
is the process for checking out the camera?
Editing
• Does at least 1 team member know how to use video editing software? Do we need to find online resources to
get help editing our video?
• Do we have access to film editing software through our school or local library, or do we need to download a free
film editing program?
• Are we going to add any music, images or video clips to our video? Do we have a way to make sure we are not
using copyrighted material in our video?
• Are we going to do voiceovers for our video or can you hear the audio in the filming well enough without it? Do
we need to add captions?
STEP 5: CREATE A STORYBOARD FOR YOUR VIDEO. A storyboard is the outline of your video’s plot, sceneby-scene. It includes drawings and/or descriptions of the basic action of your video.
Storyboarding is a really useful step to do before writing the script. It helps ensure the
script is not scattered or missing important plot points. It also can help your students
decide whether they want to have in-person acting, animations or a combination of
both in their video. Students should include the following information in their storyboard:
•
•
Who the characters are
Where the action takes place
•
•
What happens in the video
How the math problem is solved
This playbook contains a storyboarding template on pg. 51 that you and your students may want to use, but it also
may be helpful to have a large board with post-its and do a single storyboard that is large enough for all of the team
to see and contribute to. You also can go to videochallenge.mathcounts.org/storyboard for a short video that
explains how to storyboard.
STEP 6: WRITE THE SCRIPT FOR YOUR VIDEO. Every video needs a script! Even if your students are master
improvisors, having a script will help ensure that the video makes sense and explains the real-world application and
solution to the math problem.
The script should describe the setting, action
of the characters, the dialogue and any superimposed words or images (for example,
graphics that explain the solution). It also may
help to divide the script into scenes or acts. A
sample script is included at the left.
Remind students the maximum time
limit on their video is 5 minutes.
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2015-2016 MATHCOUNTS Playbook
STEP 7: DO A TIMED TABLE-READ AND RUN-THROUGH TO PRACTICE. It’s important that students have
the opportunity to hear how their script sounds when read aloud before they film anything.
A table-read is exactly what it sounds like: rather than acting out the parts, the actors simply
sit around a table and read the script from beginning to end—hopefully with the emotion and
emphasis they have planned for the final product.
Have the students time their table-read so they can ensure their final product is no more than 5 minutes in length. If
reading the script aloud takes significantly more than 5 minutes, then the students will likely need to cut some scenes
or trim the dialogue. On the other hand, if the table-read only lasts a few minutes, the students may want to consider
adding to their script to improve their final product.
If students are going to be filming anything in-person, they should do a run-through to practice their acting and dialogue. It may be a good idea for the students to perform their script for others and get feedback before filming. If the
students are making a video that is entirely animated, then a run-through is not necessary.
STEP 8: SHOOT YOUR VIDEO. The Math Video Challenge accepts a number of formats for videos, meaning your
students can use lots of different recording devices and editing programs when creating their video. Most videos
submitted to the contest are filmed with a camera, camcorder or smartphone.
It’s important to emphasize to your students that mistakes are OK! A big part of the editing process will include deleting scenes that didn’t work or adding a voiceover to the video to make it
easier to understand. Your students may need more than a few takes—and possibly more time
than they originally planned for—in order to get their video where they want it to be. If students
are planning to do in-camera editing (meaning all of the scenes of their video are shot in order),
they should keep this in mind when they are filming so they can minimize the amount of editing needed.
Here are some common pitfalls your students should try to avoid when filming:
•
•
•
Pitfall: It’s difficult to understand what is said in the video. Students must enunciate and speak loudly
during filming. It would be a good idea to have someone listen to their video who is not familiar with their script
(the team advisor, a parent, etc.), so they can identify any spots where the dialogue is unclear.
Pitfall: Background noise interferes with dialogue. Wind and other students talking can be distracting or
even make the dialogue in a video incomprehensible. No matter where students are filming, they should try to
minimize background noise. This may mean they have to reschedule a filming session, reshoot a scene or adjust
the audio during editing.
Pitfall: Filming is too far away or blurry. Students should carefully consider the placement of their camera, so
the action of the video can be seen clearly and the actors are able to face the camera when speaking. The filming
quality can also be affected by the size and resolution of the video file.
Avoiding these common mistakes during filming will help your students avoid frustration during the editing process
and make their video even better!
STEP 9: EDIT YOUR VIDEO AND ADD DESIRED SPECIAL EFFECTS. Students can opt to do in-camera
editing during filming, in order to minimize the amount of time needed for the editing process. For in-camera editing,
students must film all of the scenes of their video in order with no overlapping scenes or dialogue. Then, once they
finish filming, they simply need to combine all of the scenes together.
If students decide not to do in-camera editing, or if they want to add any special effects or
voiceover recordings, they will need to use video editing software. There are a number of
free software programs available, and many schools also have video editing software, such
as Windows Movie Maker or iMovie, installed on computers in the media center. Using video
editing programs allows students to create more interesting transitions between scenes;
add music, captions or images; and cut scenes or dialogue to improve their video.
2015-2016 MATHCOUNTS Playbook
11
Here are some common pitfalls your students should try to avoid when editing (you’ll notice some repetition, given
that a lot of pitfalls apply to both filming and editing):
•
•
•
•
•
Pitfall: It’s difficult to understand what is said in the video. If the dialogue cannot be heard, students
should use their video editing software either to boost the dialogue in their video or silence the audio and record
a voiceover that is clearer.
Pitfall: Background noise interferes with dialogue. If background noise (such as wind) makes the dialogue
difficult to hear, students should use their video editing software to reduce the background noise.
Pitfall: Filming is too far away or blurry. Students should make sure the size and resolution of the video file
are sufficient. A video that is too small or low-resolution will look blurry on most screens.
Pitfall: Video doesn’t use 5 minutes effectively. Students should ensure their video progresses smoothly
and uses time effectively, and a lot of this can be done in the editing stage. Issues such as a long introduction,
unnecessary transitions or lag-time in dialogue can be trimmed to ensure students have enough time for the most
important scenes in their video. Putting together a thorough storyboard and script, as well as doing a table-read,
can help with this.
Pitfall: Video includes copyrighted video, images or music. This is the most common mistake students
make! There are some exceptions, but generally speaking, students should not use any songs or videos they hear
on the radio, buy on iTunes or see in movies. The safest approach is for students to write their own music and
lyrics (and they’ll probably impress the judges with their creativity!). But there are other resources (listed below)
students can use to ensure they do not include audio or video material that is against the rules of the contest.
It is really important that your students’ video not include any copyrighted material. When you are
reviewing your students’ video so you can approve it, it is a good idea to watch it with this pitfall in mind. Unfortunately, including copyrighted material in the video will mean your students’ video will not be able to advance in
the contest or win any awards.
MATHCOUNTS provides a number of resources for editing and avoiding the pitfalls noted above. Here are online
resources your students can use:
General Information About Making a Video:
videochallenge.mathcounts.org/making-video
How to Make a Complete Movie
with Only an iPhone:
videochallenge.mathcounts.org/iphone
What Music and/or Video
Can I Use in My Video?
videochallenge.mathcounts.org/music
A Beginner’s Guide to iMovie:
videochallenge.mathcounts.org/imovieguide
Windows Movie Maker:
videochallenge.mathcounts.org/livemoviemaker
iMovie Support:
videochallenge.mathcounts.org/imoviesupport
STEP 10: SHOW THE FINAL PRODUCT TO SOMEONE WHO HASN’T SEEN THE
VIDEO BEFORE. Your students may be completely sure that their video is ready for the contest, but often people miss mistakes when reviewing or editing their own work. An unclear part
of dialogue, a missing plot point or a typo may be overlooked by your students because they
know what should be shown or said. Before your students submit the video for your review
and approval, have them screen it for at least 1 person who has not seen the video before.
Based on that person’s feedback, the students should make any necessary changes to their
video before submitting it.
Good luck in the challenge! If you have any questions during the year, please contact
the MATHCOUNTS national office at (703) 299-9006 or [email protected].
12
2015-2016 MATHCOUNTS Playbook
2015-16 Program Materials
Thank you for being a team advisor in the Math Video Challenge this year!
We hope participating in the program is meaningful and enriching for you and your students.
What’s in This Year’s Playbook
Critical 2015–2016 Dates .......................................................................................... 14
The 2015-2016 School Handbook .......................................................................15
250 creative math problems organized by math topic
Answers to Handbook Problems.................................................................................39
available to the general public...including your students
Problem Index + Common Core State Standards Mapping .............................. 41
all 250 problems are categorized and mapped to the CCSS
Helpful Resources for the Math Video Challenge........................................44
awesome tools and materials for team advisors and students
MATHCOUNTS Solve-A-Thon ......................................................................................... 44
Interactive MATHCOUNTS Platform ................................................................................ 45
Offline Permission Form ....................................................................................................... 46
Assessment Tools for Evaluating Videos .......................................................................... 47
Graphic Organizers for Brainstorming ............................................................................. 49
Template for Storyboarding ................................................................................................ 51
Other MATHCOUNTS Programs .............................................................................. 52
the National Math Club and MATHCOUNTS Competition Series
Registration Form for the National Math Club ........................................................ 53
Registration Form for the Competition Series ........................................................ 54
2015-2016 MATHCOUNTS Playbook
13
Critical 2015-2016 Dates
2015
September –
December
Program registration is open. Students begin working on their video(s) using this year’s
handbook problems, which are included in this 2015-2016 Playbook.
Team advisors must begin the registration process for this program online at
videochallenge.mathcounts.org/register. After the team advisor registers, s/he must
get parent/guardian approval for each student on the team. Parent approval can be done
online or team advisors can provide parents the Offline Registration Form to complete. This
form is included in this playbook on pg. 46. Any offine registration forms should be mailed
or emailed to:
MATHCOUNTS Foundation – Math Video Challenge Registrations
1420 King Street, Alexandria, VA 22314
Email: [email protected]
Questions? Call the MATHCOUNTS national office at (703) 299-9006.
2016
14
January –
March
Program registration is open. Students continue working on their video(s) using this year’s
handbook problems, which are included in this 2015-2016 Playbook.
Feb. 3
Suggested Deadline for Video Submissions
In order for your students to have the opportunity to collect votes during the entire general
public voting period, their video(s) must be submitted (and approved by you, the team advisor) by February 3.
Feb. 4 – Mar.
14
General Public Voting
Voting is limited to 1 vote per video per day for each email account used to log into the
contest website. Encourage your students to vote every day for their video(s) and ask their
friends and family members to vote daily, as well.
Mar. 14
Final Deadline for Video Submissions
Videos submitted after 11:59 PM ET on March 14 will not be eligible to win. Videos submitted on or right before the Final Deadline are unlikely to earn enough votes to advance to
the judging rounds. Get your students’ video(s) submitted and approved by February 3 to
maximize your students’ time to collect votes.
Mar. 17
Top 100 Videos Announced
Mar. 28
20 Semifinalist Videos Announced
Apr. 4
4 Finalist Videos Announced
May 8-9
2016 Raytheon MATHCOUNTS National Competition in Washington, DC
Teams that created the 4 finalist videos receive an all-expenses-paid trip to present their
work to the 224 national competitors.
2015-2016 MATHCOUNTS Playbook
The 2015-2016 School Handbook
Start here to choose a math problem to solve! We have reformatted the 2015-2016 MATHCOUNTS School
Handbook exclusively for the Math Video Challenge. In this playbook, all 250 of the handbook problems are organized
by math topic.
Because problems are organized by math topic, the problem numbers are not in sequential order. We’ve
written the problem number to the left of each problem. You can find the answer key on pgs. 39-40 and a complete
Problem Index, including difficulty level and mapping to the Common Core State Standards, on pg. 41.
If your team would prefer to use the version of the handbook formatted for the Competition Series (divided into
Warm-Ups, Workouts and Stretches), you can go to www.mathcounts.org/handbook to download a free copy of that
version. Here is where you can find this year’s handbook problems in this playbook.
Algebraic Expressions
& Equations
pgs. 16-18
Al
Coordinate Geometry
pg. 18
Cg
Gm
General Math
pg. 19
Logic
pg. 20
Lo
Me
Measurement
pgs. 20-21
!
2015-2016 MATHCOUNTS Playbook
Nt
Pf
Pg
Pc
Ps
Number Theory
pgs. 22-24
Percents & Fractions
pg. 25
Plane Geometry
pgs. 26-29
Probability, Counting
& Combinatorics
pgs. 30-32
Pr
Sp
Sg
Sd
Proportional
Reasoning
pg. 35
Sequences, Series
& Patterns
pg. 36
Solid Geometry
pgs. 37-38
Statistics & Data
pg. 38
Problem Solving
(Misc.)
pgs. 33-34
These playbook problems are not in sequential order.
They’re based on the 2015-2016 MATHCOUNTS School Handbook,
and we’ve put the problem number to the left of each math problem.
You can find the answers on pg. 39.
15
Al
Algebraic Expressions & Equations
These playbook problems are not in sequential order.
Get the answers on pg. 39.
Problem #
8
!
If 3x + 155 = 272, then what is the value of 3x + 160?
10 One-half of three-fourths of a number is 20 more than two-fifths of the same number. What is the number?
23 If
34
35
1
1
1
1
+
+
+
= 24, what is the value of x? Express your answer as a common fraction.
x
x
x
x
The price P, in dollars, that AppCo charges annual subscribers for downloading x smartphone applications
is calculated using the formula P = 5 + 1.25x. If a subscriber downloaded 25 applications during a year,
what was the average cost per application?
The local gaming club has more than 2 but fewer than 20 members. Every member of the group purchased
a signed copy of the latest book from the legendary Dumas, for a grand total of exactly $299. If each member paid the same whole number dollar amount, how many members are in the club?
38 The line that contains (4, −7) and (−3, 14) also contains the point (0, b). What is the value of b?
43
If four hamburgers and two hot dogs cost $16.40, and six hamburgers and four hot dogs cost $26.40, what
is the combined cost of a hamburger and a hot dog?
To cater a party, Fancy Caterers charges a certain amount per guest plus a fixed delivery fee. The total cost
57 for a party with 25 guests is twice the cost of a party with 10 guests. If the cost of a party with n guests is
twice the cost of a party with 25 guests, what is the value of n?
61
Zoey has $2.90 in quarters, dimes and nickels. The number of nickels is one more than six times the number
of quarters, and the number of dimes is four times the number of quarters. What is the total number of coins
Zoey has?
71
When an integer is doubled and increased by 3, the result is 5 less than the square of the integer. What is
the sum of all such integers?
Laree notices that the current mileage on her car is a multiple of 1000 and is a perfect cube. She does
74 some calculations and determines that she will have to drive another 4921 miles before the number of miles
on her car is a perfect cube again. What is Laree’s current mileage?
86
Margo is currently twice as old as Joy was three decades ago, and Joy is currently three times as old as
Margo was two decades ago. How old was Joy when Margo was born?
99 If f ( x ) = 3x − 4 and g ( x ) = x 2 − 2x + 5, what is the value of g (−4) + f (10)?
X
108 If X # Y = Y + XY, what is the value of 15 # (6 # 2)?
112 If a ▲ b = |a – b|, then what is the sum of all numbers x such that (3 ▲ x) ▲ 8 = 2?
122
16
The sum of two numbers is 7 and their difference is 18. What is their product? Express your answer as a
decimal to the nearest hundredth.
2015-2016 MATHCOUNTS Playbook
Problem #
On her 13th birthday, Moesha’s height is exactly 62 inches. If she uses the formula H(t) = 61 + 20.4t to de125 termine her height H, in inches, t years after her 13th birthday, what should Moesha expect her height to be
on her 18th birthday?
128
Age
2 & under
3 to 12
62 & older
All others
Fee
No charge
$3 each
$7 each
$10 each
On their vacation, the Ship family took a boat ride to Shipwreck Island. The
table shows the fees charged by the boat company, based on each passenger’s
age. The total charge for the 12 family members was $73. If the number who are
“others” exceeds the number who are 62 and older, what is the maximum possible number of children ages 3 to 12?
Erik and Alicia went to a restaurant and ordered the same thing for lunch. At the end of the meal, they used
134 different methods to calculate the tip. Erik paid a tip equal to twice the sales tax, and Alicia doubled the final
amount on the bill, including the sales tax, and then moved the decimal point in the resulting amount one
place to the left. Amazingly, these two methods resulted in the same tip. What was the percent sales tax?
Express your answer to the nearest tenth.
140 If a  b = a · b + 3, what is the absolute difference between (10  11)  12 and 10  (11  12)?
141 Let f(x) = 3x2 − 7 and g(x) = 2x + 5. What is the absolute difference between f(g(−2)) and g(f(−2))?
The perimeter of square A is 28 feet less than the perimeter of square B. The area of square A is 161
147 square feet less than the area of square B. If x and y are the side lengths of square A and square B, respectively, what is the value of x + y?
In a certain game, each move consists of pairing tiles of equal value to create a new tile with double the
148 value. For example, a pair of 4-tiles combine to make an 8-tile. Given an unlimited supply of 2-tiles, what is
the minimum number of pairings needed to build a tile with a value of 32?
The graph of f ( x ) =
− x 2 + 8 x − 15 contains points in how many quadrants of the Cartesian coordinate
168 plane?
5
5
If y = - x + 3 and y = - x + 11 are equidistant from a point with coordinates (t, 14), what is the value of
4
4
171
t? Express your answer as a decimal to the nearest tenth.
x
k
182 If f(x) = x and f (f (f (2))) = 2 , what is the integer value of k?
3
Lines m and n both pass through the point (2, 3) as shown. Line m has slope ,
195
4
1
and line n has slope . If lines m and n intersect the line x = 5 at the points A and B,
5
respectively, what is the distance AB? Express your answer as a common fraction.
196 What integer is closest in value to
200
1
102 - 98
y
(2, 3)
m
A
n
B
x=5
x
?
In a quadratic equation Ax2 + Bx + C = 0, A, B and C are integers whose only common factor is 1. The
2
roots of the equation are and 4. If A > 0, what is the value of A + B + C?
3
2
x
if x is even

x
−
3
?
203 What is the value of f (f (f (19) + 1) + 1) if f (x) = 
if x is odd

 2
2015-2016 MATHCOUNTS Playbook
17
Problem #
206 What is the sum of the coefficients of the terms when (2x + y)5 is expanded?
x
5x
211 If 3 = y 2, and 9 = y z, what is the value of z?
Coordinate Geometry
Cg
These playbook problems are not in sequential order.
Get the answers on pg. 39.
Problem #
154
!
Let O be the point (0, 0) in the coordinate plane, and let A be the point (3, 7). If B is the point obtained by
rotating A 90 degrees counterclockwise about O, then what is the area of triangle ABO?
y
158
The area of the region bounded by the parabola y = 4x − x2 and the line y = −5 can be
2
determined using the formula A = bh, where b is the length of the horizontal base and h
3
is the vertical distance from the vertex of the parabola to the base. What is the area of this
x
region, shown shaded?’
y
(1, 3)
177
(3, 1)
x
(–3, –1)
(–1, –3)
The figure shows a square with vertices (1, 1), (−1, 1), (−1, −1) And (1, −1) and a
curve made from four quarter-circular arcs, each centered at a vertex of the square. If
the endpoints of the four arcs are (3, 1), (1, 3), (−3, −1) and (−1, −3), what is the
area inside the curve? Express your answer in terms of π.
209 P(−3, −2) is reflected over the line y = −x and then translated 4 units right and 1 unit down. What are the
coordinates of the final image of P? Express your answer as an ordered pair.
y
4
210
The points A(4, 0), B(0, 3), C(c, 0), D(0, d) and E(e, 0) are graphed on the
B
coordinate plane as shown. If AB ^ BC, BC ^ CD and CD ^ DE, what is the 2
C
E
A
2
−2
−2
D
4
x
combined length of the four segments? Express your answer as a common
fraction.
217 In this grid composed of 16 unit squares, the five dots can be labeled with the letters A
through E so that AB < BC < CD < DE. What is the length of segment CE?
18
2015-2016 MATHCOUNTS Playbook
General Math
Gm
These playbook problems are not in sequential order.
Get the answers on pg. 39.
Problem #
!
1
If Franco types MATHCOUNTS 50 times in total, how many more consonants does he type than vowels?
2
Vicki and Candace start a race at the same time. Vicki finishes in 28 minutes 47 seconds. Candace finishes
in 29 minutes 46 seconds. How many seconds ahead of Candace does Vicki finish?
14 On a number line, what number is two-thirds of the distance from one-half to 2.25? Express your answer as
a common fraction.
22 The Population Reference Bureau reported a net gain of 155 people on Earth each minute. At this rate, how
many more people are there on Earth every day?
73 If the 15 teams in a soccer league each play eight games in a season, what is the total number of games
played during the season?
Mrs. Lowe is buying lunch for her class of 25 students. A large pizza that serves three people costs $8, and a
80 giant sub that serves four people costs $9. If pizzas and subs cannot be purchased in part, what is the least
amount it will cost Mrs. Lowe to feed all the students in her class?
A car service charges $2.40 for the first quarter of a mile traveled and then charges $0.40 for each additional
82 fifth of a mile. How many miles can a rider travel for $10? Express your answer as a decimal to the nearest
hundredth.
Laurie had 30 coins, all dimes and quarters, in a jar. Every morning for five days, she took out a quarter; then
91 she replaced it with a dime when she got home from school. After the five days, Laurie had twice as many
dimes as quarters. How many quarters were initially in Laurie’s jar?
A ball is released from a height of 16 feet, and each time it strikes the ground, it bounces back to a height
105 one-fourth of its previous height. At the instant when the ball strikes the ground for the fifth time, how many
total feet has it traveled since it was released? Express your answer as a mixed number.
In the country of Woodington, three denominations of coins are used to pay for goods and services. Each coin
110 has a value of 1 dollar, 3 dollars or 7 dollars. How many different combinations of coins can be used to pay
exactly 15 dollars in Woodington?
At 8:00 a.m., the chairperson of the homecoming committee shared this year’s theme with the three other
111 committee members. Within an hour, each of those three committee members told three people who were not
on the committee. Every hour after that, each person who had just been told within the previous hour then told
three other people who had not yet been told. How many people knew the homecoming theme by the time
the first lunch period started at 11:00 a.m.?
138
The table shows the charges associated with shipping 1 to 10 items for one particular retailer. Based on this,
what is the least amount a customer can pay per item for shipping?
Items
Shipping
1
$1.85
2
$2.15
3
$2.65
4
$3.35
5
$4.25
6
$5.35
7
$6.65
8
$8.15
9
$9.85
10
$11.75
2
2
155 How many ordered pairs of integers (x, y) satisfy the equation x + y = 2500?
Two 12-hour clocks were started simultaneously, both showing the same time. Clock A loses 15 minutes each
202 hour, and clock B gains 15 minutes each hour. How many hours pass before the first instance when Clock A
again shows the same time as Clock B?
2015-2016 MATHCOUNTS Playbook
19
Lo
Logic
These playbook problems are not in sequential order.
Get the answers on pg. 39.
Problem #
21
!
A shift consists of rotating a two-digit number 90 degrees clockwise and then exchanging the position of the
two digits. How many shifts must be performed before the number 19 returns to its original orientation?
In the figure, each circle is a vertex of one or more triangles. The circles marked B
53 and R are colored blue and red, respectively. If each of the remaining circles is to be
colored red, blue or yellow so that no triangle has two vertices of the same color, what
is the color of the circle marked with ?
B
R
Ron and Martin are playing a game with a bowl containing 39 marbles. Each player takes turns removing 1,
78 2, 3 or 4 marbles from the bowl. The person who removes the last marble loses. If Ron takes the first turn to
start the game, how many marbles should he remove to guarantee he is the winner?
1
97 If the pattern continues, what is the value of x in the third figure?
9
7
8
135
5
−3
12 2
0 6
7
x
The four vertices of a rectangle also are vertices of a regular hexagon of side length 1 unit. What is the area
of the rectangle? Express your answer in simplest radical form.
152
If this pattern continues, how many cubes will be in the next figure?
What is the maximum number of the L-pieces shown that can be placed en169 tirely on this 8 × 8 board, with none overlapping, such that the shaded sections
of each L-piece are on shaded squares of the board? (Rotating pieces and
L-piece
reflecting pieces are permitted.)
Me
Problem #
Measurement
These playbook problems are not in sequential order.
Get the answers on pg. 39.
!
Ms. Swift wrote 40 pages of notes, each page containing approximately 240 words. Chanel, who types 36
29 words per minute, volunteered to type some of Ms. Swift’s notes, beginning at the first page and progressing
through them in order. Marcus, who types 54 words per minute, volunteered to type some of Ms. Swift’s notes,
beginning at the last page and progressing through them in reverse order. If Chanel and Marcus begin at
the same time, working together, how many minutes will it take them to finish typing the notes? Express your
answer to the nearest whole number.
20
2015-2016 MATHCOUNTS Playbook
Problem #
31 On average, a human fingernail grows at a rate of 3 mm per month. At this rate, how many total meters will
Deshawn’s 10 fingernails grow over the next 10 years? Express your answer as a decimal to the nearest tenth.
The first and second books of a book series have 10 and 11 chapters, respectively, and each chapter has
37 exactly 20 pages. The total thickness of the pages in the first book is 14 mm. For book two, the total thickness
of the pages plus the front and back covers is 17 mm. The thickness of each page in book one is the same
as that in book two, and the thickness of each cover in book one is the same as that of book two. What is
the total thickness when the two books are stacked cover to cover? Express your answer as a decimal to the
nearest tenth.
59
Kaci’s jogging speed is 60% faster than her walking speed. The time it takes her to walk a mile is 5
1
minutes
2
longer than the time it takes her to jog a mile. How many seconds does it take Kaci to jog a mile?
Two lines parallel to the sides of a large rectangle divide the rectangle into four regions. The
75 areas of three of the regions, starting in the upper right corner and going counterclockwise, are
24, 40 and 15 units2 as shown (not to scale). What is the area of the large rectangle?
40
24
15
Asanji and Allen are hired to paint the fence around a neighbor’s property. Working alone, Allen can paint the
95 entire fence in 9 hours, and Asanji can paint the same fence, working alone, in 7 hours. Working together,
how many minutes will it take them to paint two-thirds of the fence? Express your answer as a decimal to the
nearest tenth.
Ted and Fred are 60 miles apart and moving toward each other. A carrier pigeon flies back and forth between
143 Ted and Fred without stopping until they meet. If Ted and Fred each maintain a constant speed of 30 mi/h
and the pigeon maintains a constant speed of 45 mi/h, what is the total number of miles the pigeon will have
flown when Ted and Fred meet?
Justin and Shelby board an escalator as it is moving down. Justin walks down 30 steps and reaches the bottom
170 in 72 seconds, while Shelby walks down 40 steps and reaches the bottom in 60 seconds. If the escalator
weren’t moving, how many steps would Justin and Shelby each have to walk down to reach the bottom?
194 If three machines can fill 80 boxes in 2 hours, how many minutes will it take five machines to fill 150 boxes?
201
1
minutes to run each mile. If she wants
3
to run the entire race in exactly 1 hour, how many minutes, on average, must she take to run each mile for the
For the first 4 miles of a 6-mile race, it took Jenny an average of 10
remainder of the race? Express your answer as a mixed number.
Alfonso, Benjamin and Carmen walk at speeds of 2, 6 and 8 feet per second, respectively. The three friends
219 start walking together at the same time in the same direction around an oval track that measures 440 yards
around. After how many minutes are the three friends next at the same exact location on the track at the same
exact time?
2015-2016 MATHCOUNTS Playbook
21
Nt
Number Theory
These playbook problems are not in sequential order.
Get the answers on pg. 39.
Problem #
9
!
Ash’s secret number is a factor of 72. The secret number is neither prime nor a multiple of 3. What is the sum
of all possible values of Ash’s secret number?
32 What percent of the prime numbers less than 100 have a units digit of 3?
What is the greatest multiple of 10 that can be expressed using only the numbers 2, 2, 3 and 5, each exactly
39 once, with any or all of the operations +, −, × and ÷, exponentiation and parentheses? Express your answer
in scientific notation.
51 What is the smallest integer that is greater than the product 2.4999 × 3.9999 × 4.9999?
A restaurant offers an incentive by giving a $5 coupon for every $40 spent during a calendar year. If Roberto
64 spends $639 at the restaurant in a year, what is the total dollar value of the coupons he can expect to receive?
68 If today is Friday, what day of the week will it be 101 days from today?
Thomas learned to count in a base other than 10. Instead of writing 163, which is in base 10, Thomas writes
69 431. What base is Thomas using?
81 How many terms of the sequence 31, 32, 33, …, 3100 have a units digit of 7?
a
x
= . If 9 is the geometric mean of m
93
x
b
and m + 3.3, what is the value of m? Express your answer as a decimal to the nearest tenth.
Two positive numbers a and b have geometric mean x if x > 0 and
113 How many positive integers m are there such that the least common multiple of m and 150 is 300?
119
Two numbers, x and y, each between 0 and 1, are multiplied. If the tenths digit of x is 1 and the tenths digit of
y is 2, what is the greatest possible value of the hundredths digit of the product?
129
What is the greatest possible sum of the reciprocals of two positive integers with a sum of 11? Express your
answer as a common fraction.
130 How many positive integers less than or equal to 100 have the same number of odd factors as even factors?
131
If a, b and c are integers between −10 and 10 inclusive such that a3 + b3 = c3, what is the greatest possible
value of a + b + c?
144 For 12 base x, written 12x, what is the value of x if 12x = 3(113)?
2240
If nk is the kth digit to the right of the decimal point in the decimal representation of
, what is the value
165
1111
of 1000n16 + 100n13 + 10n10 + n7?
166 What is the greatest prime factor of 65 – 45 – 25?
The digits 1 through 7 are each used once to write three prime numbers. Two of these primes have two digits
185 each and one has three digits. What is the greatest possible value of the three-digit prime number?
22
2015-2016 MATHCOUNTS Playbook
Problem #
213
Namakshi wants to arrange her marching band in a rectangular array with the same number of band members
in each row. But when she tried putting them in 3 rows, she had 1 band member left over; in 5 rows, she had
2 members left over; in 7 rows, she had 3 members left over; and in 9 rows, she had 4 members left over.
What is the least number of members that could be in the band?
What is the probability that a randomly chosen two-digit number containing at least one 1 or 9 is prime? Ex218 press your answer as a common fraction.
220
For each positive integer n ≤ 55, if S(n) is the sum of the positive integer divisors of n, what is the greatest
possible value of S(n)?
MODULAR ARITHMETIC STRETCH
Modular arithmetic is a system of integer arithmetic that enables us obtain information and draw conclusions
about large quantities and calculations. It would be extremely helpful, for instance, when asked to find the units
digit of 22015 if we didn’t really have to calculate the value of the expression to get that information. Modular
arithmetic allows us to do just that!
The Basics:
The simplest example of modular arithmetic is commonly referred to as “clock arithmetic.” Suppose it is 3
o’clock now and I want to know what time it will be in 145 hours. We could count from 3 o’clock for 145
consecutive hours. We certainly wouldn’t be expected to count 145 hours starting with 3 o’clock. Suppose
we did counting the hours from 3 o’clock. What happens when we get to 12 o’clock? We continue counting
but begin a new 12-hour cycle. Instead of counting 145 hours, we can just see how many of these 12-hour
cycles we’d go through counting 145 hours. More importantly, we need to determine how many hours would
remain after making it through the last full 12-hour cycle.
In this example, the value 12 is called the modulus and what is left over is called the remainder.
In this case, we can determine fairly quickly that there are 12 full 12-hour cycles in 145 hours, with a remainder
of 1 hour (since 12 × 12 = 144 and 145 − 144 = 1).
Standard arithmetic: 145 = 12 × 12 + 1
Modular arithmetic we write: 145 º 1 (mod 12) The remainder of 1 tells me that it will be the same time 145 hours after 3 o’clock that it will be 1 hour after 3
o’clock. And that time is 4 o’clock.
Here’s another example of modular arithmetic. Suppose today is Tuesday. What day of the week will it be 417
days from now? Since the days of the week are on a 7-day repeating cycle, the modulus here is 7. If we divide
417 by 7, we get
Standard arithmetic: 417 = 59 × 7 + 4
Modular arithmetic we write: 417 º 4 (mod 7)
Thus, 417 days from Tuesday will be the same day of the week as 4 days from Tuesday, Saturday.
Try these:
241 If the current month is July, what month will it be in 152 months?
242 If the time is currently 8 a.m., what time will it be in 255 hours?
243
Jennie goes out every morning and jogs on the school track. The track is 400 meters around. If Jennie runs
5310 meters then how far will she be from where she started once she finished her run?
2015-2016 MATHCOUNTS Playbook
23
MODULAR ARITHMETIC STRETCH (continued)
Modular Addition: What is the remainder when 9813 + 7762 + 11252 is divided by 10?
9813 + 7762 + 11252 = (981 × 10 + 3) + (776 × 10 + 2) + (1125 × 10 + 2)
= (981 + 776 + 1125) × 10 + (3 + 2 + 2)
Since we are only interested in the remainder, we need only focus on the last part. We see that the remainder is 3 + 2 + 2 = 7. Written in modular arithmetic notation it would look like this:
9813 +7761 +11252 º 3 + 2 + 2 º 7 (mod 10)
Modular Multiplication: What is the remainder whsen 9813 × 7762 is divided by 10?
9813 × 7762 = (981 × 10 + 3) × (776 × 10 + 2)
= (981 × 776 × 102) + (981 × 2 × 10) + (776 × 3 × 10) + (3 × 2)
The first three terms are multiples of 10, and once again last term is the remainder 3 × 2 = 6. Written in
modular arithmetic notation would look like this:
9813 × 7762 º 3 × 2 º 6 (mod 10)
More Mod Shortcuts: There are many useful applications of modular arithmetic. Here are just a few more.
•
Consider the powers of 3: 30 = 1; 31 = 3; 32 = 9; 33 = 27; 34 = 81; 35 = 243; 36 = 729
Notice that the units digits are repeated every four powers of 3, so the modulus is 4. Repeating units
digits correspond to remainders 1, 2, 3 and 0.
•
Suppose you want the unit digit of 353. First, we note that 53 º 1 (mod 4) since the remainder 1 corresponds to units digit 3, thus, the expansion of 353 has a units digit of 3.
• The smallest number that has remainder 1 when divided by 2 and 3 is 7. Why?
1 º 7 (mod 2) and 1 º 7 (mod 3)
Try these:
Problem #
244 What is the last digit of 22015?
245 What is the value of 122 × 71 modulo 11?
246 What is the remainder when 5981 × 8162 × 476 is divided by 5?
247 Jon has 29 boxes of donuts with 51 donuts in each box. He wants to divide them into groups of a dozen
each. Once he groups them again, how many donuts will be left over?
248 What is the least integer greater than 6 that leaves a remainder of 6 when it is divided by 7 and by 11?
249
250
24
When organizing her pencils, Faith notices that when she puts them in groups of 3, 4, 5, or 6, she always
has exactly one pencil left over. If Faith has between 10 and 100 pencils, how many pencils does she have?
When organizing her pens, Faith notices that when she puts them in groups of 3, 4, 5, or 6, she is always
one pen short of being able to make full groups. If Faith has between 10 and 100 pens, how many pens
does she have?
2015-2016 MATHCOUNTS Playbook
Pf
Problem #
16
Percents & Fractions
These playbook problems are not in sequential order.
Get the answers on pg. 39.
!
An item with an original price of d dollars has its price increased by x percent in April and then decreased by
x percent in May. The resulting price is 4 percent less than the original price. What is the value of x?
3
to a number increases it by 5% of its original value. What is the original number? Express your
27
8
answer as a decimal to the nearest tenth.
Adding
46
77
83
100
121
136
172
176
193
212
Kim created decorations for the school dance. It took her 4 minutes to create the first decoration, and each
decoration after the first one took 10% less time to create than the one before it. How many minutes did it take
her to create the first five decorations? Express your answer as a decimal to the nearest tenth.
The effectiveness of Donovan’s cold medication decreases geometrically, retaining one-fourth of its original
effectiveness after four hours. If Donovan takes 500 mg of medication every four hours beginning at 8:00 a.m.,
how much effective medication remains in his body at 6:00 p.m.? Express your answer as a mixed number.
Billie is downloading a video game to her computer. When she checks at 11:15 a.m., her computer indicates
that the download is 35% complete. When she checks again at 11:30 a.m., the download is 80% complete.
Assuming the video game is downloaded at a constant rate, Billie should expect the download to be complete
after an additional a minutes b seconds. If a and b are whole numbers and b < 60, what is the value of a × b?
The amount of water in a reservoir declines 10% each year. What is the minimum number of whole years it will
take until less than half of the original amount of water remains?
A photo measures 5 inches wide by 3 inches high. If the total area of the photo is enlarged by 150%, while
maintaining the same ratio of width to height, what is the height of the enlarged photo? Express your answer
as a decimal to the nearest tenth.
In a certain state, gas prices increased from $1.72 to $3.84 per gallon over a 10-year period. If the prices
increased uniformly by the same percent from year to year, what was the annual percent increase? Express
your answer to the nearest tenth.
Priya needs to purchase a camera and a printer. Priya has two coupons, one for 15% off a single item and the
other for 20% off any other single item. Priya determines that applying the 20% discount to the printer and the
15% discount to the camera would save her $0.99 more than if she applied the 15% coupon to the printer
and the 20% discount to the camera. What is the absolute difference between the price of the camera and
the price of the printer?
A math class is 60% girls. After g girls and 1 boy join the class, it is 75% girls. What is the least possible
value of g?
Kenyatta has some quarters on a table, each showing either heads or tails. Kenyatta chooses 20% of the
quarters that show heads and 10% of the quarters that show tails and then turns over each of the chosen
quarters one time. After she does this, exactly half of the quarters on the table show heads. Before Kenyatta
turned the quarters over, what was the ratio of the number of quarters that showed heads to the number that
showed tails? Express your answer as a common fraction.
What is the value of the sum 3 + 6 + 12 + 15 + 21 + 24 + + 48 + 51? Express your answer as a common frac9
18
27
54
tion.
2015-2016 MATHCOUNTS Playbook
25
Pg
Plane Geometry
These playbook problems are not in sequential order.
Get the answers on pgs. 39-40.
Problem #
3
!
What is the sum of the degree measures of the interior angles of a regular octagon?
13 What is the radius of a circle with a circumference measuring 24π cm?
15 The length of a rectangular sports field is three times its width. If the perimeter of the field is 880 meters, what
is the area of the field?
Y
19
Arc XYZ, shown here, is a semicircle. If XZ = 15 units, what is the value of
(XY)2 + (YZ)2?
Z
X
40 Two sides of a triangle are 15 cm and 18 cm in length. The altitude to the 18-cm side is 10 cm. What is the
length of the altitude to the 15-cm side?
41
Point A is a vertex of a regular octagon. When all possible diagonals are drawn from point A in the polygon,
how many triangles are formed?
47 Lily made a circle from a length of string measuring 26 cm, and Willow made a circle from a length of string
measuring 31 cm. What is the absolute difference in the radii of their circles? Express your answer as a decimal to the nearest tenth.
1
1
52 What is the area of a rectangle with sides that measure 2 2 and 7 3, in meters? Express your answer as a
mixed number.
A
58 ABCD is a trapezoid with bases AB = 18 cm and DC = 22 cm. The measures of
ABC and ADC are 120 degrees and 60 degrees, respectively. What is the perim60º
eter of trapezoid ABCD?
D
18
120º
22
B
C
70 Starting at the origin of a coordinate plane, an ant crawls 1 unit to the right, 2 units up, 3 units to the right, 4
units up, 5 units to the right and 6 units up. How far from the origin is the ant currently located?
72 Triangle ABC is isosceles with AB = AC and mA = 30 degrees. Side AB is extended to D so that
mACD = 90 degrees. What is the degree measure of BCD?
85
A newly minted half-dollar coin has a diameter of 1.205 inches and 150 equally spaced ridges around its
circumference. What is the distance between any two adjacent ridges? Express your answer as a decimal to
the nearest thousandth.
87 What is the area of a rhombus with side length 10 inches and diagonal lengths that differ by 4 inches?
88
26
Octavio bought a very unpredictable stock. Its value increased by 10% each day on Monday, Tuesday and
Wednesday, was unchanged on Thursday, then dropped by 30% on Friday. Octavio mistakenly thought that
the 30% loss was offset completely by the three 10% gains. By what percent did the stock in fact decrease
for the week? Express your answer to the nearest hundredth.
2015-2016 MATHCOUNTS Playbook
Problem #
A
N
45°
W
E
60°
B
96 Airplane A flies 45 degrees east of north at a constant speed of 300 mi/h. Traveling at
the same altitude as airplane A, airplane B flies 60 degrees west of south at a constant
speed of 250 mi/h. Both flights originate at the same time and location. What is the
distance between airplanes A and B after they have been flying for 2 hours? Express
your answer to the nearest whole number.
S
98
The largest circular pizza ever made had a diameter of 122 feet 8 inches. To divide the pizza among 150
contest winners, the chef made straight cuts through the center of the pizza, all equal in length to its diameter.
If all the slices had the same area, how many inches were there between slices, measured along the pizza’s
circumference? Express your answer as a decimal to the nearest tenth.
102
Segments of how many distinct positive lengths can be drawn using pairs of points in this
5 × 5 evenly spaced grid as endpoints?
107
In triangle ABC with acute angle BAC, the lengths of sides AB and BC are 15 units and 16 units, respectively.
How many possible integer lengths are there for side AC?
18
A
120
Quadrilateral ABCD, shown here, is a 12 × 18 rectangle. Segments BF and AG bisect
12
angles ABC and BAD, respectively, and intersect at E. What is the area of DEFG?
D
123
8 cm
B
E
F
G
C
A circle is inscribed in an isosceles right triangle with hypotenuse 8 cm. What is the
area of the circle? Express your answer as a decimal to the nearest tenth.
A
133
In Figure 1, the vertices of equilateral triangle ABC are connected with segments AB and AC. B
C
Figure 1
In Figure 2, the vertices are connected by congruent segments AD, BD and CD that intersect
at D. D is the intersection of the medians of triangle ABC. If triangle ABC has side length 1
A
unit, what is the absolute difference between the sum of the lengths of the two segments in
Figure 1 and the sum of the lengths of the three segments in Figure 2? Express your answer in
D
simplest radical form.
B
C
Figure 2
j
A
1
137
B
k
2
D
139
The figure shows parallelogram ABCD with A of measure 57 degrees. Line j is perpendicular to line k and also intersects side AB such that m1 = 81 degrees.
What is the m2?
C
The side lengths of this rectangle are in the ratio 5:7. Two isosceles right triangles are
drawn in the rectangle’s interior. The hypotenuse of each of these right triangles is one of
the longer sides of the rectangle. The shaded region represents what percent of the area of
the rectangle? Express your answer to the nearest tenth.
2015-2016 MATHCOUNTS Playbook
27
Problem #
146 Equilateral triangle ABC has side length 6 units, and M is the midpoint of altitude AD. The
a b
length of segment MC expressed as a common fraction in simplest radical form is
c
units, where a, b and c are integers. What is the value of a + b + c?
160
A
M
B
C
D
An 8-inch by 12-inch paper napkin is folded in half three times, with each fold resulting in a smaller rectangle. What is the longest possible diagonal for the final rectangle? Express your answer in simplest radical
form.
161
A right triangle has sides, in inches, measuring 2n, n2 − 1 and n2 + 1, where n is a positive integer. What is
the area of this triangle if its perimeter is 40 inches?
164
The area of a rectangle is 168 in2, and its perimeter is 62 inches. What is the product of the lengths of the
diagonals of this rectangle?
Square ABCD, shown here, has side length 4 inches. The arc from A to B is part of a circle
167
whose center coincides with the center of the square, point E. What is the area of the entire
figure? Express your answer in terms of π.
174
178
B
E
D
C
The figure shows a quarter of unit circle P and a quarter of unit circle Q drawn with
the center of each on the circumference of the other. What is the area of the shaded
region? Express your answer as a decimal to the nearest hundredth.
P
A
Q
This octagon is classifed as isogonal since each of its vertices is between one short side and
one long side, and its interior angles are all congruent. If the long sides have length 120 feet
and the distance between two parallel long sides is 190 feet, what is the perimeter of the
octagon? Express your answer to the nearest whole number.
190 ft
120 ft
183
184
Circle O, with radius 8 units, and circle W, with radius 3 units, are externally tangent. The circles are also
tangent to line m at distinct points B and L, respectively. What is the area of quadrilateral BOWL? Express
your answer in simplest radical form.
E
In triangle DEF, DE = 6 cm, DF = 9 cm and DG = 7 cm, where G is the midpoint of
side EF. What is the length of side EF? Express your answer in simplest radical form.
G
D
F
AREA STRETCH
231
28
Norm has a square sheet of paper with 10-inch sides. Along each side, he makes a
mark 2 inches from each corner. He then draws a line segment connecting the two
marks near each corner. Finally, he cuts along each line segment, removing a triangle
from each corner of the square and creating an octagon. What percentage of the area
of the square is the area of the octagon?
10 in.
2 in.
2015-2016 MATHCOUNTS Playbook
AREA STRETCH (continued)
11 ft.
Problem #
232
The figure shows an office floor plan. How many square feet does this office occupy?
15 ft.
12 ft.
24 ft.
233
234
235
236
A running track consists of two parallel straight segments, each 100 meters long, connected by two semicircular stretches, each with inner diameter 50 meters. What is the total area enclosed by the running track?
Express your answer to the nearest hundred.
What is the greatest possible area of a concave pentagon in the coordinate plane with vertices (−2, 0),
(2, 0), (2, 10), (0, 6) and (−2, 10)?
A square is inscribed in a circle of radius 4 units. The square divides the interior of the
circle into five regions, four of which lie outside the square. What is the area of the shaded
region? Express your answer in terms of π.
Amy marks two points A and B that are 4 inches apart. She draws one circle that has segment AB as a
diameter. She then draws a larger circle, which overlaps the first circle, such that the arc from A to B along
its circumference is a quarter-circle. What is the total area covered by the two circles? Express your answer
in terms of π.
A
237
B
C
H
D
G
F
E
In convex octagon ABCDEFGH, shown here, each side has length 6 units, and diagonals
AE and CG have length 16 units. If the octagon is symmetric across both diagonals AE
and CG, what is its area? Express your answer in simplest radical
form.
E
238
In this figure, AE = EQ = BC = CP = 10 units, and AQ = BP = 12 units. The
points A, P, Q and B are collinear. If the perimeter of the concave pentagon ABCDE is 52 units, what is its area? Express your answer as a common fraction.
B
239 A
P
Q
B
Right triangle ABC with AC = 3 units, BC = 4 units and AB = 5 units is rotated
90° counterclockwise about M, the midpoint of side AB, to create a new right triangle
A’B’C’. What is the area of the shaded region where triangles ABC and A’B’C’ overC' lap? Express your answer as a common fraction.
B'
240
D
A'
M
C
C
A
In right triangle ABC, C is a right angle, AC = 10 units and BC = 24 units. If a point X is located inside triangle ABC so that the distance from X to side AB is twice the distance from X to side AC, and the distance
from X to side AC is twice the distance from X to side BC, what is the distance from X to side AB? Express
your answer as a common fraction.
2015-2016 MATHCOUNTS Playbook
29
Pc
Probability, Counting & Combinatorics
These playbook problems are not in sequential order.
Get the answers on pg. 40.
!
Problem #
5
Lily is going to the movies with Abby, Bea and Jaclyn. Abby wants to sit at the end of a row, and Bea only
cares that she is seated next to Jaclyn. In how many different ways can the girls be seated in a single row that
has only four seats?
20 Seven contestants enter a drawing that begins with 100 balls numbered 1 through 100 in a box. Each contestant randomly selects a ball without replacement. The two contestants who select balls with the two highest
numbers each will win a cash prize. The first six contestants select balls numbered 83, 5, 44, 67, 21 and 30.
What is the probability that the last contestant will win a cash prize? Express your answer as a common fraction.
In the 6 × 6 array of squares shown, two squares are adjacent if they share a side. What is the
25 probability that two adjacent squares, chosen at random, are not the same color? Express your
answer as a common fraction.
26
A very round number is a positive integer that has exactly one nonzero digit. How many integers less than one
billion are very round numbers?
At Grace Hopper Middle School, there are 531 students in the sixth, seventh and eighth grades combined.
The table below shows the number of students in each grade who play soccer and the number of students
44
in each grade who do not play soccer. If all of the seventh and eighth graders attend an assembly, and two
students at the assembly are chosen at random, what is the probability, as a percent, that neither student plays
soccer? Express your answer to the nearest tenth.
49
55
Grade
Play Soccer
Don’t Play Soccer
6
79
118
7
61
116
8
54
103
In how many ways can the integers 1 through 6 be written horizontally in a row so that the sum of any two
adjacent integers is odd?
What is the minimum number of people that must be in a room to guarantee that at least one person has a
birthday whose day is a single digit, assuming that no two people have the same birthday?
Every day, a cafeteria offers pizza, chicken and salad as lunch entrées. This comes with a choice of milk, juice,
water or tea to drink and a cookie or brownie for dessert. If a different combination of one entrée, one drink
62
and one dessert is tried every day, how many days does it take to try every combination?
67
Two distinct numbers are selected at random from the set {1, 2, 3, 4, 5, 6}. What is the probability that their
product is an odd number? Express your answer as a common fraction.
On average, one out of every 70 widgets manufactured has a defect. If Nick inspects four widgets chosen at
random, what is the probability, expessed as a percent, that at least one of them will be defective? Express
84
your answer to the nearest tenth.
30
2015-2016 MATHCOUNTS Playbook
Problem #
115 A standard deck of cards consists of cards numbered 2 through 10 plus a Jack, Queen, King and Ace in each
of four different suits. In a particular card game, Jacks, Queens and Kings are each worth 10 points, Aces are
worth 11 points and numbered cards are worth face value. Each of four players is dealt three cards, and the
winner is the player with the greatest sum of cards of the same suit. AUSTIN Player
BAILEY
B COOPER Player
DALLAS
D
If cards have been dealt to the four players as shown, and Austin
Ace ♦
Ace ♥
King ♣
8♦
then is dealt a card from those remaining in the deck, what is the
9♦
King ♦
3♥
9♣
probability that Austin has the winning hand? Express your answer
?
3♠
Queen ♦
5♥
as a common fraction.
117
In how many ways can each of the digits 3, 5 and 7 be used exactly once to replace X, Y and Z to make the
true inequality 0.XY < 0.Z?
127 A
B
1 2 3 4 5 6
The figure shows a rectangle divided into 12 congruent squares. In how many
ways can each square be colored red, blue or green, so that when the rectangle
is rotated 180° every location (for example, row A column 1) contains a square
of a different color than the one that was originally in that location?
132 The positive integers 1 through 50 are written on 50 cards with one integer on each card. If Matt draws one
card at random, what is the probability that the number on the card is a multiple of 6 or 8? Express your
answer as a common fraction.
145 How many different quadrilaterals have vertices with integer coordinates (x, y) such that 0 ≤ x ≤ 2 and
0 ≤ y ≤ 2?
151
For each of the 20 basketball games this season, Coach Washington needs to choose 5 players to start. If
he doesn’t want the same 5 players starting together more than once, what is the minimum number of players the coach needs on his team roster?
Spirit Committee members will be chosen from students nominated in each grade. Two students will be
157 chosen from the five nominated 6th graders. Four students will be chosen from the seven nominated 7th
graders. Six students will be chosen from the nine nominated 8th graders. In how many different ways can
the committee members be chosen?
173 The Lemurs are playing the Capybaras in a series of three baseball games. The games alternate between
the Lemurs’ home field and the Capybaras’ home field, with the first game taking place at the Lemurs’ field.
Each team has a 60% chance of winning a game on its home field. What is the probability, expressed as a
percent, that the Lemurs will win at least two of the three games? Express your answer to the nearest tenth.
180
Will and Ian each randomly choose a positive divisor of 20. What is the probability that the least common
multiple of their chosen numbers is 20? Express your answer as a common fraction.
181 How many positive integer divisors does 2016 have?
191
199
When three fair dice are rolled, what is the probability that the product of the three numbers rolled is a prime
number? Express your answer as a common fraction.
Four husband-and-wife couples attend a show. From these four couples, two people are randomly selected when the performer asks for volunteers from the audience. What is the probability that the two who are
selected are a married couple? Express your answer as a common fraction.
208 How many 4-digit integers have their digits in strictly ascending order?
2015-2016 MATHCOUNTS Playbook
31
COUNTING STRETCH
Problem #
221 Hazel wrote the integers 1 through 321 on the board. How many total digits did she write?
222 How many triangles of any size are in this figure?
223
In how many ways can one knife, one fork and one spoon be distributed, in any order, to three people, if each
person is given 0, 1, 2 or 3 utensils?
224 Using pennies, nickels, dimes and quarters, how many ways can you make 67 cents?
225
In the game Fortrix, a player can earn 3, 7 or 11 points on a turn. How many different scores are possible for
a single player after six turns?
226 How many 3-digit integers are divisible by both 5 and 17?
227 How many positive integers less than 40 are relatively prime to both 7 and 10?
228 How many palindromes are between 9 and 1009?
R O R
229
In the 3 × 3 grid shown, a path can begin in any cell and can pass through a cell more than once.
How many such paths spell ROTOR?
O T
O
R O R
H
T
230
Moving only up and right, how many paths from P to H pass through A and T?
A
P
32
2015-2016 MATHCOUNTS Playbook
Problem Solving (Misc.)
Ps
These playbook problems are not in sequential order.
Get the answers on pg. 40.
Problem #
7
!
What is the maximum number of non-overlapping regions that can be determined by three lines in a plane?
11 What is the least possible sum of the digits displaying the time on a 12-hour digital clock?
12 How many 4-digit palindromes contain both the digits 1 and 2?
G A B
48 The digits of the addends in the sum shown are represented by the letters A, B and G. What + B A G
is the value of A × (B + G)?
1 0 9 0
66
101
1
2 3
4 5
6
This figure, composed of six congruent squares labeled 1 through 6, is a net of a cube. If it is
folded to form a cube, what is the sum of the digits on the four faces that are adjacent to the face
labeled 1?
In a talent contest involving three finalists, audience members vote for first, second and third place by assigning
each a 1, 2 or 3, respectively. When all the votes have been totaled, the finalist with the lowest total wins. If
100 audience members vote and there are no ties in the outcome, what is the greatest number of points that
the winner can get?
1 1
If l, m and n are each distinct members of the set {2, - , , −3, 12}, what is the least possible value of
116
6 3
l×
´m
?
n
126
When filled in correctly with nonzero digits, each row and column in this cross-number grid contains a perfect square. No two rows contain the same perfect square, and no two columns contain
the same perfect square. What is the least possible sum of all the digits in the completed grid?
A positive integer k is said to be divisive if k > 10, all digits of k are nonzero and each digit of k,
except the units digit, is a proper divisor (any factor of the number except the number itself) of the digit to its
150
immediate right. Based on this, how many positive integers are divisive?
Each letter of the alphabet is assigned a numerical value equal to its position, so that A = 1, B = 2,
C = 3, …, Z = 26. The value of a word is the sum of the values of its letters. For example, the value of THREE
156
is 20 + 8 + 18 + 5 + 5 = 56. What integer from 1 to 20, inclusive, when spelled out, has the greatest value?
In the country of Fizzle, coins come in denominations of 5, 8 and 11 fizz. What is the greatest integer amount
159 that cannot be paid with a whole number of these coins?
How many positive three-digit integers have the property that the tens digit is the sum of the units digit and
162 the hundreds digit?
Four rows from a sheet of isometric dot paper, where the rows contain three, two, three and two
points, respectively, are shown. How many distinct non-equilateral triangles can be drawn with each
187
vertex at a point in this array?
2015-2016 MATHCOUNTS Playbook
33
Problem #
188
Two positive integers not exceeding 100 have the property that their sum is divisible by their difference. What
is the greatest possible value of their difference?
189 If Pascal’s Triangle is folded along its vertical line of symmetry and then unfolded, what is the sum of the seven
numbers on the fold line that are of least value?
190 The figure shows part of an infinite tree diagram in which each fraca
a
a+b
tion
has the fractions
and
written below it, starting
b
a+b
b
1
at Level 1 with . For what integer n does Level n contain the frac1
31
tion
?
4159
1
1
Level 1:
1
3
Level 3:
Level 4:
2
1
1
2
Level 2:
1
4
3
2
4
3
3
5
3
1
2
3
5
2
2
5
5
3
3
4
4
1
197 Victor pays $1.00 for a bottle of soda at his local market. Each time Victor returns three empty bottles, he earns
a free bottle of soda. What is the minimum amount Victor must spend to get 10 bottles of soda?
198
207
216
34
The figure shows a regular hexagon and seven circles, one at each vertex and another at the
hexagon’s center. Three diagonals are drawn through the center of the hexagon, connecting
opposite vertices. In how many ways can each of the circles be colored either red or blue so
that no three collinear circles are the same color?
In a certain game of darts, a dart that lands on the bull’s-eye scores 11 points, and a dart that lands any other
place on the board scores 7 points. A player throws three darts, and the probability that the player hits the
bull’s-eye is 50% on each throw, independent of previous results. Given that each dart lands on the board,
and at least one dart lands on the bull’s-eye, what is the probability that the total number of points scored will
be prime? Express your answer as a common fraction.
Stacey has some pennies, nickels and dimes divided between her left and right pockets. She has the same
number of coins in each pocket, and the coins in her two pockets add to the same total value. Stacey does
not have the same number of pennies in each pocket. What is the minimum total value that she could have in
both pockets combined?
2015-2016 MATHCOUNTS Playbook
Pr
Proportional Reasoning
These playbook problems are not in sequential order.
Get the answers on pg. 40.
!
Problem #
4
Vera’s favorite coffee blend costs $1.32 per ounce, including tax. How much will it cost Jorge to buy Vera one
and a half pounds of her favorite coffee?
18
Josh mowed one-third of a 2000-ft2 lawn in 18 minutes. At the same rate, how many minutes would it take
him to mow a 4200-ft2 lawn? Express your answer to the nearest whole number.
36
42
Jamee is building a two-story house in which the top floor is 12 feet above the bottom floor.
The local building code specifies that the height of a step cannot exceed 7.5 inches. What is
the maximum height of a step in a staircase that Jamee can construct between the two floors
if all steps will have the same height? Express your answer as a decimal to the nearest tenth.
12'
Alicia is framing a rectangular photo that measures 8 inches high by 12 inches wide. However, the frame she has
was designed to hold a picture with sides in the ratio 4:5. If the frame Alicia has is the smallest possible frame
that holds the photo, how much extra area will there be? Express your answer as a decimal to the nearest tenth.
A cable car is 30 feet in length and travels back and forth in a straight line, on a single cable that is 3000 feet
in length. Let P be a point somewhere on the cable car. During one round trip,
56
what is the distance traveled by point P?
Thirty percent of 80% of 20 is the same as what percent of 60% of 40?
60
On Oscar’s trip to school, he took 50 minutes to walk 2.7 miles. He walked at a constant speed for the first
40 minutes, then increased his speed by exactly 1 mi/h. How fast was Oscar walking for the first 40 minutes
94 of his trip? Express your answer as a decimal to the nearest hundredth.
The time it takes a cube of ice to melt at a certain temperature varies directly with the surface area of the cube. If
it takes 3 hours for a 1-foot cube of ice to melt, how many minutes will it take a 1-inch cube of ice to melt if
103 the temperature remains constant? Express your answer as a decimal to the nearest hundredth.
A passenger jet flies a certain distance in 3 hours 30 minutes when traveling west to east. The same distance
from east to west requires 4 hours 15 minutes. What is the ratio of the passenger jet’s speed going east to
106 the speed going west? Express your answer as a common fraction.
109
When Jose looked at the clock, he noticed that 20% of the total time from 4:00 p.m. to 5:00 p.m. had elapsed.
What percent of the time from 1:00 p.m. to 6:00 p.m. had elapsed, when Jose looked at the clock?
A tree is 8 feet tall when planted. After one year, its height has increased by one-sixth of its original height. At
the end of the second year, its height is 37.5% greater than its original height. How many inches did the tree
192 grow during the second year?
Rectangle ABCD, shown here, has point E on side AB such that the ratio of the area of trape- B
zoid EBCD to that of triangle AED is 5:1. What is the value of the ratio AE:EB? Express your
204
answer as a common fraction.
C
E
A
2015-2016 MATHCOUNTS Playbook
D
35
Sequences, Series & Patterns
Sp
These playbook problems are not in sequential order.
Get the answers on pg. 40.
Problem #
Letter Ms are stacked, as shown, so that the top row has one M, the second row has two
17
Ms, the third row has three Ms, and so on. What is the least number of rows required for the
total number of stacked Ms to be divisible by 7?
!
M
M M
M M M
M M M M
24 Maximo wrote down two rows of numbers as shown. In the first row, he wrote the positive multiples of 7,
starting with 7 and ending with 700. In the second row, he wrote the positive multiples of 13, starting with
13 and ending with 1300. How many times did corresponding pairs of terms in the first and second rows
have the same units digit?
First row:
7
14
21
28
35
…
700
Second row:
1326395265… 1300
30
76
An exponentially talented salesman sells 1 car on his first day, 2 cars on his second day, 4 cars on his third
day and so on, so that every day after the first, he sells twice as many cars as the day before. How many
days does it take him to sell a total of at least 1000 cars?
The first three terms of an arithmetic sequence are 17x + 20, 18x – 3 and 20x + 1, in that order. What is
the value of x?
79 What common fraction is equal to the sum
114
1
1
1
1
+
+
+ ... +
?
2
4
8
512
Carrick is learning to read. During his first lesson, Carrick read 50 words, and during each lesson thereafter,
he read 10 more words than he read during the lesson before. If Carrick had one reading lesson each day
for 30 days, what is the total number of words he read in 30 lessons?
142 What is the value of 1 – 2 + 3 – 4 + 5 – 6 + ... + 2013 − 2014 + 2015?
153
In a particular arithmetic sequence, the fourth term is 38, and the sum of the second and seventh terms is
85. What is the value of the fifth term of the sequence?
An arithmetic sequence of positive integers has a common difference that is three times the first term. If the
163 sum of the first five terms is equal to the absolute difference between the first term and its square, what is
the first term in the sequence?
175 For a particular sequence, if a1 = 2 and an + 1 = −an + 2n for n ≥ 1, what is a2016?
All terms of an arithmetic sequence are integers. If the first term is 13, the last term is 77 and the sequence
179 has n terms, what is the median of all possible values of n?
In a certain sequence of numbers, every term after the second term is equal to the sum of the two preceding
215 terms. If the 99th term of the sequence is 7 and the 102nd term is 69, what is the 100th term of the sequence?
36
2015-2016 MATHCOUNTS Playbook
Solid Geometry
Sg
These playbook problems are not in sequential order.
Get the answers on pg. 40.
!
Problem #
28
45
What is the height of a right rectangular prism with a length of 4 cm, a width of 3 cm and a volume of
108 cm3?
It costs $0.12 for the Cheapco Soda Company to manufacture enough cola to fill a cylindrical container
that is 2 inches in diameter and 4 inches in height. How much would it cost for Cheapco to manufacture
enough cola to fill a cylindrical container that is 6 inches in diameter and 10 inches in height?
50 What is the radius of the largest sphere that will fit inside a cube of volume 8 units3?
63
65
A pile of clay can be molded to form a solid cube with edges of length 10 cm. How many solid rectangular
prisms with dimensions 2 × 4 × 5 cm can be made from the same amount of clay?
GG’s shipping company ships golf balls in three different box sizes. The small box contains a dozen golf
balls. The medium box is similar to the small box, but each dimension is doubled. The large box also is similar
to the small box, but each dimension is tripled. If Jake orders one box of each size, how many golf balls will
he receive?
89 Tootie Frootie candy is cylindrical in shape and comes in two different sizes. One size is 1.25 inches long
with a circumference of 2 inches, and the other is 3.25 inches long and has twice the volume of the first.
What is the circumference of the larger size? Express your answer as a decimal to the nearest hundredth.
90 A 32-foot by 15-foot by 8-foot hole is being dug in Winnie’s yard for a rectangular swimming pool. If the
truck hauling away the dirt holds a maximum of 6 cubic yards, how many truckloads of dirt will be taken
away? Express your answer as a whole number.
12
92 A garden has the shape of a rectangle with semicircles on either end as shown. The
length of the rectangular portion is 12 feet, and the width is 8 feet. How many cubic
yards of topsoil are needed to fill the entire garden uniformly to a depth of 2 inches?
Express your answer as a decimal to the nearest tenth.
124
186
8
A water tank in the shape of a cone, with its point down as shown, has base radius 12
feet and height 8 feet. If the tank is half full by volume. What is the ratio of the depth of the
water to the height of the tank? Express your answer as a decimal to the nearest hundredth.
Dan tilts his soda can until the soda is halfway up the can at its highest point and a third of
the way up the can at its lowest point. If the can is 4.5 inches tall with a 2.5-inch diameter,
how many cubic inches of soda are in the can? Express your answer as a decimal to the
nearest tenth.
2015-2016 MATHCOUNTS Playbook
1
3
1
2
37
Problem #
205 Kendra will make a box with an open top by cutting, folding and taping a 12-inch by 16-inch rectangular
piece of cardboard. She will begin by cutting from each corner of the flat cardboard a square with a side
length that is a multiple of 0.5 inch. What is the maximum possible volume of the box? Express your answer
as a decimal to the nearest tenth.
214
Three pyramids, each created using the net shown here, can be combined to form
a cube. What is the volume of the cube?
4 cm
Sd
Statistics & Data
These playbook problems are not in sequential order.
Get the answers on pg. 40.
Problem #
6
!
On the first five math quizzes of the school year, Maurice’s scores were 80, 84, 99, 92 and 100. After the
sixth quiz, Maurice’s median score for all six quizzes was 90. What was Maurice’s score on the sixth quiz?
33 In Mrs. Abel’s class, there are 16 girls and 14 boys. The mean of the girls’ heights is 63.7 inches, and the
mean of the boys’ heights is 65.4 inches. What is the mean of the heights of all 30 students? Express your
answer as a decimal to the nearest tenth.
54 The mean of the first five terms of an arithmetic sequence is 27, and the mean of the first eight terms of
the same sequence is 39. What is the absolute difference between the first and second terms of this sequence?
104
118
The mean score on the last test in Ms. McMean’s class of 24 students is 88. If the four lowest scores are
excluded, the mean is 94. If the four highest scores are excluded, the mean is 86. What is the absolute difference between the means of the four highest and four lowest test scores?
For a list of eight positive integers, the mean, median, unique mode and range are 8. What is the greatest
integer that could be in this set?
The absolute difference between the mean and median of five distinct positive integers is at least 2. If the
149 five integers are 3, 22, 7, 12 and x, with 3 and 22 being the least and greatest values, respectively, what is
the sum of all possible values of x?
38
2015-2016 MATHCOUNTS Playbook
Answers to Handbook Problems
Al
Cg
Lo
Nt
Pf
Algebraic
Expressions &
Equations
8. 277
10. –800
23. 1/6
34. $1.45
35. 13 members
38. 5
43. $5 or $5.00
57. 55
61. 34 coins
71. 2
74. 64,000 miles
86. 12 years old
99. 30.8
108. 226
112. 12
122. –68.75
125. 65 inches
128. 4 children
134. 11.1%
140. 6
141. 19
147. 23
148. 15 pairings
168. 3 quadrants
171. –5.6
182. 2048
195. 33/20 units
196. 5
200. –3
203. 16
206. 243
211. 20
Coordinate
Geometry
Logic
Number
Theory
Percents &
Fractions
154. 29 units2
158. 36 units2
177. 10π –4 or
–4 + 10π units2
209. (6, 2)
210. 875/64 units
217. 1 unit
21. 4 shifts
53. red
78. 3 marbles
97. 3
135. √3 units2
152. 50 cubes
169. 16 L-pieces
9. 13
32. 28%
39. 1 × 109
51. 50
64. $75 or
$75.00
68. Monday
69. base 6
81. 25 terms
93. 7.5
113. 6 integers
119. 5
129. 11/10
130. 25 integers
131. 20
144. 10
165. 2016
166. 7
185. 547
213. 157 members
218. 13/34
220. 124
241. March
242. 11:00 p.m.
243. 110 m
244. 8
245. 5
246. 2
247. 3
248. 83
249. 61
250. 59
16. 20
27. 7.5
46. 16.4 minutes
Gm
General
Math
1. 200
2. 59 seconds
14. 5/3
22. 223,200
people
73. 60 games
80. $60 or
$60.00
82. 4.05 miles
91. 15 quarters
5
105. 26 8 feet
110. 10 combinations
111. 121 people
138. $0.84
155. 20 pairs
202. 24 hours
2015-2016 MATHCOUNTS Playbook
Me
Measurement
29. 107 minutes
31. 3.6 m
37. 32.6 mm
59. 550 seconds
75. 88 units2
95. 157.5 minutes
143. 45 miles
170. 90 steps
194. 135 minutes
1
201. 9 3 minutes
219. 11 minutes
1
77. 328 8 mg
83. 240
100. 7 years
121. 4.7 inches
136. 8.4%
172. $19.80
176. 6
193. 4/3
212. 191/20
Pg
Plane
Geometry
3. 1080 degrees
13. 12 cm
15. 36,300 m2
19. 225 units2
40. 12 cm
41. 6 triangles
47. 0.8 cm
52. 18 13 m2
58. 48 cm
70. 15 units
72. 15 degrees
39
Pg
Pc
Ps
Pr
Sg
Plane
Geometry
(continued)
85. 0.025 inches
87. 96 in2
88. 6.83%
96. 1091 miles
98. 30.8 inches
102. 14 lengths
107. 25 lengths
120. 9 units2
123. 8.6 cm2
133. 2 – √3 or
–√3 + 2 units
137. 48 degrees
139. 5.7%
146. 12
160. √145 inches
161. 60 in2
164. 625 in2
167. 2π + 12 or
12 + 2π in2
174. 0.17 units2
178. 678 feet
183. 22√6units2
184. √38 cm
231. 92%
232. 321 ft2
233. 7000 m2
234. 32 units2
235. 4π – 8 or
–8 + 4π units2
236. 4 + 8π or
8π + 4 in2
237. 128 + 32√2
or 32√2 + 128
units2
238. 357/4 units2
239. 9/4 units2
240. 240/37
units
Probability,
Counting &
Combinatorics
5. 8 ways
20. 16/47
25. 1/3
26. 81 integers
44. 42.9%
49. 72 ways
55. 259 people
62. 24 days
67. 1/5
84. 5.6%
115. 5/41
117. 3 ways
127. 46,656
ways
132. 6/25
145. 78 quadrilaterals
151. 7 players
157. 29,400
ways
173. 55.2%
180. 5/12
181. 36 divisors
191. 1/24
199. 1/7
208. 126 integers
221. 855
222. 56 triangles
223. 27 ways
224. 87 ways
225. 13 scores
226. 10 integers
227. 14 integers
228. 100 palindromes
229. 64 paths
230. 54 paths
Problem
Solving
(Misc.)
7. 7 regions
11. 1
12. 2 palindromes
48. 40
66. 15
101. 199 points
116. –144
126. 26
150. 22 integers
156. 17
159. 17 fizz
162. 45 integers
187. 93 triangles
188. 66
189. 1275
190. 145
197. $7 or $7.00
198. 54 ways
207. 3/7
216. 90 cents
Proportional
Reasoning
Solid
Geometry
40
4. $31.68
18. 113 minutes
36. 7.2 inches
42. 19.2 in2
56. 5940 feet
60. 20%
94. 3.04 mi/h
103. 1.25 minutes
106. 17/14
109. 64%
192. 20 inches
204. 1/2
28. 9 cm
45. $2.70
50. 1 unit
63. 25 prisms
65. 432 golf balls
89. 1.75 inches
90. 24 truckloads
92. 0.9 yd3
124. 0.79
186. 9.2 in3
205. 192.5 in3
214. 64 cm3
Sd
Sp
Sequences,
Series &
Patterns
17. 6 rows
24. 20 times
30. 10 days
76. –27
79. 511/512
114. 5850 words
142. 1008
153. 47
163. 36
175. 2014
179. 9 terms
215. 31
Statistics &
Data
6. 88
33. 64.5 inches
54. 8
104. 40
118. 14
149. 23
2015-2016 MATHCOUNTS Playbook
Problem Index + Common Core
State Standards Mapping
Problems Mapped to Common Core State Standards
Forty-three states, the District of Columbia, four territories and the Department of Defense Education Activity (DoDEA) have adopted the
Common Core State Standards (CCSS). As such, MATHCOUNTS considers it beneficial for teachers to see the connections between the
2015-2016 MATHCOUNTS School Handbook problems and the CCSS. MATHCOUNTS not only has identified a general topic and assigned
a difficulty level for each problem but also has provided a CCSS code in the Problem Index (pgs. 42-43). A complete list of the Common Core
State Standards can be found at www.corestandards.org.
The CCSS for mathematics cover K-8 and high school courses. MATHCOUNTS problems are written to align with
the NCTM Standards for Grades 6-8. As one would expect, there is great overlap between the two sets of standards.
MATHCOUNTS also recognizes that in many school districts, algebra and geometry are taught in middle school, so
some MATHCOUNTS problems also require skills taught in those courses.
In referring to the CCSS, the Problem Index code for each of the Standards for Mathematical Content for grades K-8
begins with the grade level. For the Standards for Mathematical Content for high school courses (such as algebra or
geometry), each code begins with a letter to indicate the course name. The second part of each code indicates the
domain within the grade level or course. Finally, the number of the individual standard within that domain follows. Here
are two examples:
• 6.RP.3 → Standard #3 in the Ratios and Proportional Relationships domain of grade 6
• G-SRT.6 → Standard #6 in the Similarity, Right Triangles and Trigonometry domain of Geometry
Some math concepts utilized in MATHCOUNTS problems are not specifically mentioned in the CCSS. Two examples
are the Fundamental Counting Principle (FCP) and special right triangles. In cases like these, if a related standard could
be identified, a code for that standard was used. For example, problems using the FCP were coded 7.SP.8, S-CP.8
or S-CP.9 depending on the context of the problem; SP → Statistics and Probability (the domain), S → Statistics and
Probability (the course) and CP → Conditional Probability and the Rules of Probability. Problems based on special
right triangles were given the code G-SRT.5 or G-SRT.6, explained above.
There are some MATHCOUNTS problems that either are based on math concepts outside the scope of the CCSS or
based on concepts in the standards for grades K-5 but are obviously more difficult than a grade K-5 problem. When
appropriate, these problems were given the code SMP for Standards for Mathematical Practice. The CCSS include the
Standards for Mathematical Practice along with the Standards for Mathematical Content. The SMPs are (1) Make sense
of problems and persevere in solving them; (2) Reason abstractly and quantitatively; (3) Construct viable arguments
and critique the reasoning of others; (4) Model with mathematics; (5) Use appropriate tools strategically; (6) Attend
to precision; (7) Look for and make use of structure and (8) Look for and express regularity in repeated reasoning.
Problem Index
In addition to the Common Core Mapping, we have provided a difficulty rating for each problem. Our scale is 1-7,
with 7 being the most difficult. These are only approximations, and how difficult a problem is for a particular student
will vary. Below is a general guide to the ratings:
Difficulty 1/2/3 – One concept; one- to two-step solition; appropriate for students just starting the middle school
curriculum.
4/5 – One or two concepts; multistep solution; knowledge of some middle school topics is necessary.
6/7 – Multiple an/or advanced concepts; multistep solution; knowledge of advanced middle school topics
and/or problem-solving strategies is necessary. 2015-2016 MATHCOUNTS Playbook
41
Problem Index
42
6.RP.3
6.RP.3
6.NS.1
6.RP.3
7.NS.3
7.RP.3
6.EE.7
6.RP.3
6.RP.3
6.RP.3
6.RP.3
7.RP.2
9 (3) 4.OA.4
32 (2) 7.NS.2
39 (4) 8.EE.2
51 (2) 7.NS.3
64 (4) 7.NS.3
68 (3) 7.NS.3
69 (4) SMP
81 (3) F-BF.2
93 (4) SMP
113 (4) SMP
119 (3) 6.NS.3
129 (3) 7.NS.3
130 (4) SMP
131 (3) SMP
144 (4) SMP
165 (5) 7.NS.2
166 (5) 8.EE.1
185 (4) SMP
213 (4) SMP
218 (4) 7.SP.7
220 (5) SMP
Modular Arithmetic Stretch*
8
(2)
6.EE.3
10
(4)
6.EE.7
23
(4)
8.EE.7
34
(5)
F-IF.2
35
(2)
A-REI.4
38
(4)
8.F.3
43
(3)
A-CED.2
57
(4)
8.EE.8
61
(4)
8.EE.8
71
(4)
A-CED.1
74
(4)
A-SSE.3
86
(5)
A-CED.2
99
(1)
F-IF.2
108
(3)
6.EE.2
112
(4)
6.EE.2
122
(4)
8.EE.8
125
(3)
F-IF.2
128
(5)
8.EE.8
134
(5)
8.EE.7
140
(3)
F-IF.2
141
(4)
8.F.1
147
(4)
A-SSE.3
148
(4)
4.OA.4
168
(4)
A-SSE.3
171
(5)
8.F.3
182
(4)
F-IF.2
195
(4)
8.F.3
196
(5)
A-SSE.2
200
(4)
A-SSE.3
203
(5)
F-IF.2
206
(3)
A-SSE.1
211
(4)
N-RN.2
Problem Solving (Misc.)
(2)
(3)
(3)
(4)
(3)
(4)
(4)
(3)
(3)
(3)
(3)
(4)
Algebraic Expressions & Equations
4
18
36
42
56
60
94
103
106
109
192
204
General Math
Number Theory
Proportional Reasoning
As noted earlier, it is difficult to categorize many of the problems in the MATHCOUNTS School Handbook. It is very
common for a MATHCOUNTS problem to straddle multiple categories and cover several concepts. This index is intended
to be a helpful resource, but since each problem has been placed in exactly one category and mapped to exactly one
Common Core State Standard (CCSS), the index is not perfect. In this index, the code 9 (3) 7.SP.3 refers to problem 9
with difficulty rating 3 mapped to CCSS 7.SP.3. For an explanation of the difficulty ratings or an explanation of the CCSS
codes refer to pg. 41.
7
11
12
48
66
101
116
126
150
156
159
162
187
188
189
190
197
198
207
216
1
2
14
22
73
80
82
91
105
110
111
138
155
202
(2)
(1)
(2)
(3)
(3)
(3)
(4)
(4)
(4)
(3)
(3)
(3)
(4)
(4)
(4)
(6)
(3)
(5)
(3)
(5)
(1)
(2)
(3)
(2)
(3)
(3)
(3)
(3)
(3)
(3)
(3)
(2)
(5)
(3)
SMP
SMP
SMP
SMP
SMP
SMP
SMP
SMP
SMP
SMP
SMP
SMP
S-CP.9
SMP
S-CP.9
SMP
SMP
SMP
SMP
SMP
4.OA.2
7.NS.3
6.NS.1
4.OA.2
4.OA.2
N-Q.1
7.NS.3
A-CED.1
7.NS.3
SMP
7.NS.3
6.NS.3
SMP
SMP
2015-2016 MATHCOUNTS Playbook
(2)
(4)
(4)
(4)
(4)
(4)
(4)
(4)
(5)
(5)
(5)
(4)
F-BF.2
F-BF.2
F-BF.2
F-BF.2
F-BF.2
F-LE.2
F-BF.2
F-LE.2
F-LE.2
F-BF.2
F-BF.2
F-BF.2
2015-2016 MATHCOUNTS Playbook
G-GMD.3
G-GMD.3
G-GMD.3
7.G.6
7.G.6
G-GMD.3
8.G.9
8.G.9
G-GMD.3
G-GMD.3
G-GMD.4
G-GMD.3
154
(4)
6.G.3
158
(5)
A-SSE.3
177
(5)
7.G.6
209
(5)
8.G.1
210
(6)
8.G.8
217
(4)
8.G.8
29
31
37
59
75
95
143
170
194
201
219
6
33
54
104
118
149
(5)
(2)
(4)
(5)
(3)
(4)
(3)
(4)
(4)
(3)
(4)
(2)
(3)
(5)
(4)
(4)
(4)
6.RP.3
6.RP.3
7.RP.3
7.RP.3
7.G.6
6.RP.3
6.RP.3
7.RP.3
6.RP.3
7.RP.3
6.RP.3
6.SP.5
6.SP.5
F-BF.2
6.SP.5
6.SP.5
6.SP.5
Logic
(3)
(4)
(3)
(5)
(5)
(5)
(3)
(4)
(5)
(5)
(4)
(3)
21
53
78
97
135
152
169
(2)
(2)
(4)
(3)
(5)
(3)
(4)
8.G.3
SMP
SMP
SMP
G-SRT.6
SMP
SMP
Percents & Fractions
Solid Geometry
Coordinate Geometry
Measurement
28
45
50
63
65
89
90
92
124
186
205
214
16
27
46
77
83
100
121
136
172
176
193
212
(5)
(4)
(3)
(4)
(4)
(4)
(3)
(5)
(4)
(5)
(5)
(4)
6.EE.7
7.RP.3
7.NS.3
7.NS.3
6.RP.3
7.NS.3
6.RP.3
7.RP.3
7.RP.3
6.RP.3
A-CED.2
7.NS.3
Probability, Counting & Combinatorics
17
24
30
76
79
114
142
153
163
175
179
215
(3) 7.G.5
(2) 7.G.4
(3) 7.G.6
(4) G-C.2
(4) 7.G.6
(3) SMP
(2) G-C.2
(2) 6.NS.1
(4) 7.G.6
(4) 8.G.8
(3) 7.G.5
(3) 7.G.4
(5) 7.G.6
(4) 7.NS.3
(5) 8.G.7
(3) 7.G.4
(3) 8.G.8
(6) 8.G.5
(3) G-SRT.5
(4) G-C.2
(5) G-SRT.6
(5) 7.G.5
(5) G-SRT.6
(5) G-SRT.6
(4) 8.G.7
(4) A-SSE.3
(5) 8.G.7
(5) G-SRT.6
(5) G-SRT.6
(4) G-SRT.6
(6) 7.G.6
(5) 8.G.7
Stretch*
Statistics
Plane Geometry
Sequences, Series & Patterns
3
13
15
19
40
41
47
52
58
70
72
85
87
88
96
98
102
107
120
123
133
137
139
146
160
161
164
167
174
178
183
184
Area
5 (3) S-CP.9
20 (4) 7.SP.8
25 (5) 7.SP.3
26 (4) S-CP.9
44 (4) 7.SP.8
49 (4) S-CP.9
55 (4) 7.SP.8
62 (3) S-CP.9
67 (4) 7.SP.8
84 (4) 7.SP.7
115 (4) 7.SP.8
117 (3) 7.SP.7
127 (5) S-CP.9
132 (4) 7.SP.8
145 (5) S-CP.9
151 (4) S-CP.9
157 (4) S-CP.9
173 (5) 7.SP.8
180 (5) 7.SP.8
181 (3) 4.OA.4
191 (4) 7.SP.8
199 (4) 7.SP.8
208 (5) S-CP.9
Counting Stretch*
43
Helpful Resources for the
Math Video Challenge
MATHCOUNTS Solve-A-Thon
Solve-A-Thon is a free fundraiser that empowers students and teachers to use math to raise money for
the math programs at their school. All money raised goes to support math programming that benefits the
students’ local communities, with 60% going directly back to the school. Launched last year, Solve-AThon was designed with teachers in mind; participating is free, and getting started takes just a couple
of minutes. Here’s how the fundraiser works:
1. Teachers and students sign up and
create online fundraising pages that
explain why they value math and why
they are raising money for their school’s
math program.
2. Students share the links to their
fundraising pages with friends, family
and local community members to earn
donations and pledges.
3. Students solve an online Problem
Pack with 20 math problems covering
topics from the NCTM Grades 6-8
Standards.
4. Students, teachers, schools and local communities win critical funding,
prizes and improved math programs.
Unlike traditional fundraisers, there is no door-to-door
selling and no tracking of sales or money raised. Students participate in a meaningful learning activity to
raise money, and money earned is tracked online automatically. Learn more and sign up at
solveathon.mathcounts.org.
44
2015-2016 MATHCOUNTS Playbook
Interactive MATHCOUNTS Platform
The Interactive MATHCOUNTS Platform provides a unique forum where members of the MATHCOUNTS community
can collaborate, chat and utilize innovative online features as they work on problems from MATHCOUNTS handbooks
and competitions.
Powered by NextThought, the Interactive
MATHCOUNTS Platform continues to grow, with
more problems and features being added every year.
This resource includes problems and step-by-step
solutions for all 250 math problems in the 20152016 MATHCOUNTS School Handbook, as well
as past handbooks and School, Chapter and State
Competitions from multiple years.
Users can take advantage of numerous features that
make this platform engaging; here are just a few:
• Digital white-boards enable students to highlight
problems, add notes and questions and show
their work.
• Interactive problems can be used to assess
student or team performance.
• Advanced search tools make it easy to find
MATHCOUNTS content and notes.
• Collaborative forums allow users to chat and
share with the global MATHCOUNTS community.
The Interactive MATHCOUNTS Platform is a great resource for your Math Video Challenge team! Create your free
account at mathcounts.nextthought.com today!
2015-2016 MATHCOUNTS Playbook
45
2015-2016 OFFLINE
PERMISSION
FORM
TEAM ADVISOR: Please give this Offline Registration Form to the parent/guardian whose name you provided
during your online registration. If you need to update this information, you can log in at the Math Video Challenge
website (http://videochallenge.mathcounts.org) and make any necessary changes online. Do not give this form to any
parents/guardians who have completed or will complete the online permission form.
PARENT/GUARDIAN: As soon as possible, please complete this Offline Registration Form and email it to
[email protected] or mail it to MATHCOUNTS – Math Video Challenge Registrations, 1420 King
Street, Alexandria, VA 22314. Your child cannot participate in the Math Video Challenge without your permission.
TEAM INFORMATION:
!
If your son/daughter is participating in more than one Math Video Challenge team, please write the names of all teams
and team advisors on this form.
(*required information)
Team Advisor Name* ________________________________________________________________________________
Team Name* _______________________________________________________________________________________
STUDENT INFORMATION:
!
If you have more than one child participating in the Math Video Challenge this year, you must complete a separate
permission form for each child.
(*required information)
First & Last Name* _______________________________________________ Grade Level (circle one)* 6
7
8
Email Address _____________________________________________________________________________________
Name of Student’s Official School of Record* ___________________________________________________________
School City* ______________________________________________________________ School State* ____________
PARENT/GUARDIAN INFORMATION:
(*required information)
First & Last Name* _______________________________________________ Phone Number* ____________________
Email Address _____________________________________________________________________________________
By signing below I attest that I am the parent/guardian of the above-mentioned minor and give permission for my
child to participate in the Math Video Challenge video contest. My child and I agree to be bound by the terms and
conditions of participation.
A copy of the terms and conditions of participation can be found at http://videochallenge.mathcounts.org/rules or can be requested by emailing
[email protected] or mailing a self-addressed stamped envelope to: MATHCOUNTS – Math Video Challenge Rules, 1420 King
Street, Alexandria, VA 22314. Please direct any questions to MATHCOUNTS at [email protected] or (703) 299-9006.
__________________________________
Printed Name of Parent/Guardian
46
__________________________________
Signature of Parent/Guardian
_______________
Date
2015-2016 MATHCOUNTS Playbook
SAMPLE ASSESSMENT TOOL
with point values pre-assigned
Name
Video Title
24 POINTS - MATHEMATICAL CONTENT
______ (0-6 pts) An appropriate approach to the solution of the problem is used.
______ (0-6 pts) The facts and logic are correct. There are no errors.
______ (0-6 pts) The use of vocabulary and notation are correct.
______ (0-6 pts) The solution is explained in a way that is clear to the viewer.
30 POINTS - COMMUNICATION
______ (0-6 pts) The mathematical thinking is communicated coherently and in a logical manner.
______ (0-6 pts) The video uses time effectively.
______ (0-6 pts) The important ideas are emphasized.
______ (0-6 pts) Sufficient detail is used.
______ (0-6 pts) The method of communication (verbal/written/etc.) is clear and easy to understand.
24 POINTS - CREATIVITY
______ (0-6 pts) The style of the video is well-suited for the audience/middle school students.
______ (0-6 pts) The story line of the video is engaging.
______ (0-6 pts) The video shows imagination by the creators.
______ (0-6 pts) The video is memorable.
10 POINTS - REAL-WORLD APPLICATION OF MATH
______ (0-10 pts) The video presents a clear, real-world application of the math concept(s) in the
problem.
12 POINTS - VIDEO LOGISTICS
______ (0-6 pts) The video is no more than 5 minutes in length.
______ (0-6 pts) The video was submitted on time.
BONUS POINTS
______ (0-2 pts) A real-world application of the concept is shown that is not part of the original
problem.
______ (0-2 pts) The video production quality is excellent.
______ (0-2 pts) The problem selected is a more difficult problem to solve and explain.
______ (0-2 pts) More than one solution to the problem is shown.
TOTAL POINTS (out of 100)
2015-2016 MATHCOUNTS Playbook
47
BLANK ASSESSMENT TOOL
without pre-determined point-values
Name
Video Title
POINTS ______ (
pts)
______ (
pts)
______ (
pts)
______ (
pts)
MATHEMATICAL CONTENT
An appropriate approach to the solution of the problem is used.
The facts and logic are correct. There are no errors.
The use of vocabulary and notation are correct.
The solution is explained in a way that is clear to the viewer.
POINTS ______ (
pts)
______ (
pts)
______ (
pts)
______ (
pts)
______ (
pts)
COMMUNICATION
The mathematical thinking is communicated coherently and in a logical manner.
The video uses time effectively.
The important ideas are emphasized.
Sufficient detail is used.
The method of communication (verbal/written/etc.) is clear and easy to understand.
POINTS ______ (
pts)
______ (
pts)
______ (
pts)
______ (
pts)
CREATIVITY
The style of the video is well-suited for the audience/middle school students.
The story line of the video is engaging.
The video shows imagination by the creators.
The video is memorable.
POINTS - REAL-WORLD APPLICATION OF MATH
______ (
pts) The video presents a clear, real-world application of the math concept(s) in the
problem.
POINTS - VIDEO LOGISTICS
______ (
pts) The video is no more than 5 minutes in length.
______ (
pts) The video was submitted on time.
BONUS POINTS
______ (
pts) A real-world application of the concept is shown that is not part of the original
problem.
______ (
pts) The video production quality is excellent.
______ (
pts) The problem selected is a more difficult problem to solve and explain.
______ (
pts) More than one solution to the problem is shown.
TOTAL POINTS (out of
48
)
2015-2016 MATHCOUNTS Playbook
s
My
a
e
video id
2015-2016 MATHCOUNTS Playbook
49
__________
__________
__________
d __________
d __________
d __________
o
o
o
e
My vide
__________ My vide
__________ My vid
__________
i
ea:
i
ea:
__________________
__________________
__________________
__________________
__________
d __________
o
e
My vid
__________
Details: _____________
__________________
__________________
__________________
__________________
__________________
__________________
__________________
__________________
__________
d __________
o
My vide
__________
Details: _____________
__________________
__________________
__________________
__________________
__________________
__________________
__________________
__________________
__________
d __________
My video
__________
Details: _____________
__________________
__________________
__________________
__________________
Details: _____________ Details: _____________ Details: _____________
i
ea:
i
ea:
i
ea:
i
ea:
50
2015-2016 MATHCOUNTS Playbook
2015-2016 MATHCOUNTS Playbook
____________________________________ ____________________________________ ____________________________________
____________________________________ ____________________________________ ____________________________________
____________________________________ ____________________________________ ____________________________________
____________________________________ ____________________________________ ____________________________________
____________________________________ ____________________________________ ____________________________________
____________________________________ ____________________________________ ____________________________________
STORYBOARD TEMPLATE
51
Other MATHCOUNTS Programs
MATHCOUNTS was founded in 1983 as a way to provide new avenues of engagement in math for middle school
students. MATHCOUNTS began solely as a competition, but has grown to include 3 unique but complementary
programs: the MATHCOUNTS Competition Series, the National Math Club and the Math Video Challenge.
Your students can participate in all 3 MATHCOUNTS programs! Below is some additional information about the
National Math Club and the MATHCOUNTS Competition Series.
The National Math Club is a free enrichment program that provides teachers and club leaders with resources to run a
math club. The materials provided through the National Math Club are designed to engage students of all ability levels—
not just the top students—and are a great supplement for classroom teaching. This program emphasizes collaboration
and provides students with an enjoyable, pressure-free atmosphere in which they can learn math at their own pace.
Active clubs also can earn rewards by having a minimum number of club members participate (based on school/
organization/group size). There is no cost to sign up for the National Math Club, and registration is open to schools,
organizations and groups that consist of at least 4 students in 6th, 7th and/or 8th grade and have regular in-person
meetings. More information can be found at www.mathcounts.org/club, and the 2015-2016 Registration Form is
included on the next page.
The MATHCOUNTS Competition Series is the longest-running MATHCOUNTS program. Established in 1983,
the Competition Series provides students the opportunity to participate in live, in-person math contests against and
alongside their peers. Students can participate in this program through their school.
Beginning with School Competitions and progressing to Chapter (local) and State Competitions, bright, motivated
students have the opportunity to hone their problem-solving skills, connect with other students who share their passion
for math and earn recognition and awards. The top 224 students from the State Competitions receive an all-expensespaid trip to the 2016 Raytheon MATHCOUNTS National Competition in Washington, DC, where they vie for the titles
of National Champion and First Place Team. More information is available at www.mathcounts.org/competition, and
the 2015-2016 Registration Form is included in this book on pg. 54.
52
2015-2016 MATHCOUNTS Playbook
!
The fastest way to register
your school is online at
www.mathcounts.org/clubreg!
2015-2016 REGISTRATION FORM
Step 1: Tell us about your group. Select and complete only one option below.
 U.S. school with students in 6th, 7th and/or 8th grade
School Name: ________________________________________________
 Chapter or member group of a larger organization
(Girl Scouts, Boys & Girls Club, nationwide tutoring centers, etc.)
Larger Organization Name: _____________________________________
There can be multiple clubs at the
same U.S. school, as long as each
club has a different club leader.
One Example:
Larger Organization Name: Girl Scouts
Chapter Name: Troop 1234
Chapter (or Equivalent) Name: ___________________________________
 A homeschool or group not affiliated with a larger organization
Club Name: __________________________________________________
Some Examples:
homeschools, neighborhood math
groups, independent tutoring centers
Step 2: Confirm that your group is eligible to participate in the National Math Club.
Please check off that all 3 of the following statements are true for your group:

My group consists of at
least 4 U.S. students.

The students in my group are
in 6th, 7th and/or 8th grade.

My group has regular inperson meetings.
By signing below I, the club leader, affirm that all of the above statements are true, to the best of my knowledge, and that my
group is therefore eligible to participate in the National Math Club. I understand that MATHCOUNTS can cancel my membership at any time if it is determined that my group is ineligible.
Club Leader Signature: ________________________________
Step 3: Provide your information so we can send you materials and set up your online access.
*required information
Club Leader Name* _______________________________________________ Club Leader Phone _______________________
Club Leader Email Address* _________________________________________________________________________________
Club Leader Alternate Email Address _________________________________________________________________________
Club Address* ____________________________________________________________________________________________
City, State ZIP* ____________________________________________________________________________________________
Estimated total number of participating students in club (minimum 4 students): ___________
 My school previously participated in the National Math Club.
School Type (please check one):
 Public
 Charter
 Private
 Homeschool
 Virtual
Department of Defense or State Department schools, please provide additional information below.
Clubs located outside of the U.S. states or territories are not eligible to participate in the National Math Club unless they are
in schools affiliated with the U.S. Department of Defense or State Department.
My school is sponsored by (please check one):
 U.S. Department of Defense (DoDDS)
 U.S. State Department
Confirm School Name: ___________________________________________ Country __________________________________
Step 4: Almost done... just turn in your form.
Mail, email a scanned copy or fax this completed form to:
MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314
Email: [email protected]
Fax: 703–299–5009
2015-2016 MATHCOUNTS Playbook
Questions?
Please call the national office at
703-299-9006
53
54
2015-2016 MATHCOUNTS Playbook
Step 1: Tell us about your school (please print legibly).
2015–2016 SCHOOL
REGISTRATION FORM
$50
$80
$60
$50
$25
$30
2
(2 ind)
1
(1 individual)
$110
$90
$75
3
(3 ind)
$120
$100
$90
4
(1 team)
$150
$130
$115
5
(1 tm, 1 ind)
$180
$160
$140
6
(1 tm, 2 ind)
$210
$190
$165
7
(1 tm, 3 ind)
$240
$220
$190
8
(1 tm, 4 ind)
$270
$250
$215
9
(1 tm, 5 ind)
$300
$280
$240
10
(1 tm, 6 ind)
!
Amount Due $________________  Check (payable to MATHCOUNTS Foundation)  Money Order  Purchase Order #_________________________ (must include copy of P.O.)
Credit card payments cannot be accepted with paper registrations. To pay with a credit card, please register online at www.mathcounts.org/compreg.
IMPORTANT! By submitting this form you (1) agree to adhere to the rules of the MATHCOUNTS Competition Series; (2) attest you have the school administraQuestions?
tion’s permission to register students for this program under this school’s name; and (3) affirm the above named school is a U.S. school eligible for this program
Please call the
and not an academic or enrichment center. The coach will receive an emailed confirmation and receipt once this registration has been processed.
national office at
Mail, email a scanned copy or fax this completed form to: MATHCOUNTS Foundation, 1420 King Street, Alexandria, VA 22314
703-299-9006.
Email: [email protected] Fax: 703-299-5009
Step 3: Almost done... just fill in payment information and turn in your form!
Principal Name ___________________________________________________________ Principal Signature ______________________________________________________________
 My school qualifies for the 50% Title I discount, so the Amount Due in Step 3 will be half the amount I circled above. Principal signature required below to verify Title I eligibility.
(postmarked after Dec. 11, 2015)
Late Registration +$20 late fee
(postmarked by Dec. 11, 2015)
Regular Rate
(postmarked by Nov. 13, 2015)
Early Bird Rate
# of Students You Are
Registering
Please circle below the number of students you will bring to the Chapter Competition and the associated cost (depending on the date your registration is postmarked).
Step 2: Sign up your students by following the instructions below.
Department of Defense or State Department Schools, please provide additional information below. Schools located outside of the U.S. states or territories are not
eligible to participate in the Competition Series unless they are affiliated with the U.S. Department of Defense or State Department.
My school is sponsored by (please check one):  U.S. Department of Defense (DoDDS).
 U.S. State Department.
Country ________________________________
Alternate Address ________________________________________________________________________________________________________________________________________
Total # of students participating in MATHCOUNTS meetings at your school* ________
 My school previously participated in the MATHCOUNTS Competition Series.
How did you hear about MATHCOUNTS?*  Mailing  Word-of-mouth  Conference  Internet  Email  Prior participation
Ship my competition materials to (please check one)*:  the School Address written above.  an Alternate Address, which I have written below.
Co-Coach Email or Coach Alternate Email ___________________________________________________________________________________________________________________
Coach Email* ____________________________________________________________________________________________________________________________________________
County/District* ___________________________________ MATHCOUNTS Chapter (if known) ______________________________ Coach Phone ____________________________
School Address* _________________________________________________________________________________________________________________________________________
School Name* _________________________________________________________________________ School Type*  Public  Charter  Private  Homeschool  Virtual
Coach Name* __________________________________________________________ Co-Coach Name (if applicable) ____________________________________________________
*required information
!
The fastest way to register is online
at www.mathcounts.org/compreg!
ACKNOWLEDGMENTS
The MATHCOUNTS Foundation wishes to acknowledge the hard work and dedication of those volunteers instrumental in
the development of this handbook: the question writers who develop the questions for the handbook and competitions, the
judges who review the competition materials and serve as arbiters at the National Competition and the proofreaders who
edit the questions selected for inclusion in the handbook and/or competitions.
2014-2015 QUESTION WRITING COMMITTEE
Chair: Edward Early, St. Edward’s University, Austin, TX
Zach Abel, Cambridge, MA
Susan Holtzapple, Los Altos, CA
John Jensen, Fountain Hills, AZ
Cody Patterson, Tucson, AZ
Carol Spice, Pace, FL
Kris Warloe, Corvallis, OR
2015-2016 NATIONAL JUDGES
Richard Case, Computer Consultant, Greenwich, CT
Flavia Colonna, George Mason University, Fairfax, VA
Barbara Currier, Greenhill School, Addison, TX
Peter Kohn, James Madison University, Harrisonburg, VA
Monica Neagoy, Mathematics Consultant, Washington, DC
Harold Reiter, University of North Carolina-Charlotte,
Charlotte, NC
Dave Sundin (STE 84), Statistics and Logistics
Consultant, San Mateo, CA
2015-2016 NATIONAL REVIEWERS
William Aldridge, Springfield, VA
Erica Arrington, North Chelmsford, MA
Sam Baethge, San Marcos, TX
Alyssa Briery (STE 04, 05, 06), Denton, TX
Lars Christensen (STE 89), St. Paul, MN
Dan Cory (NAT 84, 85), Seattle, WA
Roslyn Denny, Valencia, CA
Joyce Glatzer, Woodland Park, NJ
Choon Woo Ha (NAT 00), Boston, MA
Special Thanks to:
Mady Bauer, Bethel Park, PA
Jerrold W. Grossman, Rochester, MI
Helga Huntley (STE 91), Newark, DE
David Kung (STE 85, NAT 86), Lexington, MD
Howard Ludwig, Ocoee, FL
Tina Pateracki, Mt. Pleasant, SC
Sandra Powers, Daniel Island, SC
Randy Rogers (NAT 85), Davenport, IA
Dianna Sopala, Fair Lawn, NJ
Craig Volden (NAT 84), Earlysville, VA
Jane Lataille, Los Alamos, NM
Leon Manelis, Orlando, FL
The Solutions to the problems were written by Kent Findell, Diamond Middle School, Lexington, MA.
MathType software for handbook development was contributed by Design Science Inc., www.dessci.com, Long Beach, CA.
MATHCOUNTS FOUNDATION
Editor and Contributing Author:
Content Editor:
Author of Introduction & Program Information:
Executive Director:
Honorary Chairman:
Kera Johnson, Manager of Education
Cara Norton, Senior Education Coordinator
Amanda Naar, Communications Manager
Louis DiGioia
Thomas A. Kennedy, Chairman and CEO of Raytheon Company
©2015 MATHCOUNTS Foundation
1420 King Street, Alexandria, VA 22314
703-299-9006 ♦ www.mathcounts.org ♦ [email protected]
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.
The MATHCOUNTS Foundation makes its products and services available on a nondiscriminatory basis. MATHCOUNTS does not
discriminate on the basis of race, religion, color, creed, gender, sexual orientation, physical disability or ethnic origin.