Mathcounts 2020-2021 (2024)

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Check out this year's math problems on pg.12!

How To Use This School Handbook IF YOU’RE A NEW COACH Welcome! We’re so glad you’re a coach this year. Please read the New Program Info for 2020-21 starting on the next page, then check out the Guide for New Coaches starting on page 4. IF YOU’RE A RETURNING COACH Welcome back! Thank you for coaching again. Please read the New Program Info for 2020-21 starting on the next page, then access this year’s Handbook Materials on page 9.

NEW PROGRAM INFO FOR 2020-21 Welcome to the MATHCOUNTS® Competition Series! The safety and wellbeing of our students, coaches, volunteers and members of the MATHCOUNTS community are our top priority, which means the Competition Series needs to be a little different this year...but still an awesome experience! If you have questions about the program details below, please feel free to contact the MATHCOUNTS national office at [emailprotected]. DUE TO COVID-19, THIS YEAR’S MATHCOUNTS COMPETITION SERIES HAS BEEN MODIFIED. Created in 1983, the MATHCOUNTS Competition Series is a national program that provides students in grades 6-8 the opportunity to compete in live math contests against and alongside their peers. Typically, these competitions are conducted as in-person events, but just for this year, we have decided to conduct online competitions at each level leading up to the National Competition. HOW WILL IT WORK? The 2020-2021 Competition Series will have 4 levels of official competition—chap- ter, chapter invitational, state and national—and 4 unofficial online practice competitions. Schools register in the fall and work with students during the year. Students will have the opportunity to take 4 online practice competitions beginning in October. Any number of students from a school can participate in team meetings. Practice competitions will be released October 15, November 15, December 15 and January 22. MATHCOUNTS strongly recommends schools participate in these practice competitions. It is important your students know how to use the competition platform before competition day. Any student whose school is not participating in the program can regis- ter as a non-school competitor (NSC). MATHCOUNTS encourages students to pursue all avenues to participate through their school before registering as NSCs. Between 1 and 15 individuals from each school participate in the 2021 Chapter Competition, which will be available online February 5 at 1:00pm ET through February 6 at 1:00pm ET. Coaches determine which students will participate in the Chapter Competition. The competition will be conducted through the Art of Problem Solving (AoPS) Contest Platform, and each student will compete solely as an individual; there will be no team competition. Like school competitors, NSCs will take the Chapter Competition online through the AoPS Contest Platform February 5-6 and will compete as individuals. top 20% The top scoring student from each school and the top 20% of individuals from or top in school each chapter advance to the 2021 Chapter Invitational, taking place online February 25 at 7:00pm ET. Every competitor advancing to this level will be required to take the competition on the AoPS Contest Platform at the same time. Each student will compete solely as an individual; there will be no team competition. NSCs can only advance to the Chapter Invitational by scoring in the top 20% of their chapter. Like school competitors, NSCs must take the 2021 Chapter Invita- tional on February 25 at 7:00pm ET on the AoPS Contest Platform. 2 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Top students from the Chapter Invitational advance to the 2021 State Com- petition, taking place online March 25 at 7:00pm ET. (State level advancement 56 details will be announced prior to the Chapter Invitational.) Every competitor advancing to this level will be required to take the competition on the AoPS Contest Platform at the same time. Each student will compete solely as an individual; there will be no team competition. Advancement to the State Competition will be the same for both school compet- itors and NSCs. NSCs must take the 2021 State Competition on March 25 at 7:00pmET on the AoPS Contest Platform. The top individuals from each state receive an all-expenses-paid trip to the 2021 Raytheon Technologies MATHCOUNTS National Competition, which takes place May 8-11 in Washington, DC. Up to 224 students will be invited to compete for individual and team awards. As of August 2020, this is planned as an in-person event, but subject to change, if necessary. Advancement to the national competition will be the same for both school compet- itors and NSCs. An NSC’s state will be determined by school location. HOW WILL REGISTRATION WORK? Registration this year is a 2-step process: 1. Registering and paying through the MATHCOUNTS Foundation 2. Setting up online competition access through Art of Problem Solving (AoPS) 1 Coaches or school officials should register their schools online at www.mathcounts.org/compreg: • June 15 – October 1: Early Bird Registration ($30 per student) • October 2 – December 1: Regular Registration ($35 per student) • December 2 – January 15: Late Registration ($40 per student) Before registering, non-school competitors (NSCs) must confirm with school officials that their school will not be participating in the Competition Series. MATHCOUNTS (1) may contact the schools of registered NSCs to confirm that the school is not participating in the Competition Series and (2) reserves the right to cancel an NSC’s registration, without refund, if their school registers for the Competition Series. NSCs must register online and pay with a credit card at www.mathcounts.org/nscreg: • August 3 – October 1: Early Bird Registration ($60 per student) • October 2 – December 1: Regular Registration ($65 per student) • December 2 – January 15: Late Registration ($70 per student) Schools and NSCs must register and pay by January Competition COACHES: 15, 2021 Please be sure to complete to participate. MATHCOUNTS cannot guarantee participa- tion for registrants with outstanding invoices after January 15. both parts of the registration process 2 Because arrangements must be made for online com- petitions... this year! • All coaches, school competitors and NSCs must create a free account at AoPS. Coach- es at schools will be given instructions to ensure their students are signed up with AoPS and linked to their school. • Schools must indicate which of their students will participate in the Chapter Competition by January 15, 2021. MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 3

Guide For New Coaches Welcome to the MATHCOUNTS® Competition Series! Thank you so much for serving as a coach this year. Your work truly does make a difference in the lives of the students you mentor. We’ve created this Guide for New Coaches to help you get acquainted with the Competition Series and understand your role as a coach in this program. If you have questions at any point during the program year, please feel free to contact the MATHCOUNTS national office at [emailprotected]. WHAT DOES THE TEST LOOK LIKE? MATHCOUNTS competition typically consists of 4 rounds—Sprint, Target, Team and Countdown Rounds. Altogether the rounds take about 3 hours to complete. However, Team and Countdown Rounds will not be conducted officially in the 2020-2021 Competition Se- ries until the national level. Here’s what each round looks like. VS Sprint Round Target Round Team Round Countdown Round 40 minutes Approx. 30 minutes 20 minutes Maximum of 45 30 problems total 8 problems total 10 problems total seconds per problem no calculators used calculators used calculators used no calculators used focus on speed and focus on problem- focus on problem- focus on speed and accuracy solving and solving and accuracy mathematical reasoning collaboration National level only for The problems are given National level only for 2020-2021. to students in 4 pairs. 2020-2021. The online Students have 6 minutes practice competitions to complete each pair. will have team rounds. HOW DO I GET MY STUDENTS READY FOR THESE COMPETITIONS? What specifically you do to prepare your students will depend on your schedule as well as your students’ schedules and needs. But in general, working through lots of different MATHCOUNTS problems and completing practice competitions are the best ways to prepare. Each year, MATHCOUNTS provides the MATHCOUNTS School Handbook to all coaches, plus lots of additional free resources online. This guide will explain the layout of the School Hand- book and other resources, plus give you tips on structuring your team meetings and preparation schedule. THE ROLE OF THE COMPETITION COACH Your role as the coach is such an important one, but that doesn’t mean you need to know everything, be a math expert or treat coaching like a full-time job. Every MATHCOUNTS coach has a different coaching style and you’ll find the style that works best for you and your students. But in general, every good MATHCOUNTS coach must do the following: • Schedule and run an adequate number of practices for participating students (these can be online). • Help motivate and encourage students throughout the program year. • Select the 1-15 student(s) who will represent the school at the Chapter Competition in February. • Ensure students can log in on the AoPS Contest Platform and are familiar with using the platform. 4 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

You don’t need to know how to solve every MATHCOUNTS problem to be an effective coach. In fact, many coaches have told us that they themselves improved in mathematics through coaching. 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 coach. You can prepare your students using solely the free resources and this handbook. We give coaches numerous detailed resources and recognition materials so you can guide your Mathletes® to success even if you’re new to teaching, coaching or compe- tition math, and even if you use only the free resources MATHCOUNTS provides all competition coaches. MAKING THE MOST OF YOUR RESOURCES As the coach of a registered competition school, you already have received what we at MATHCOUNTS call the School Competition Kit. Your kit includes the following materials for coaches. 2020-2021 MATHCOUNTS Student Ribbons and School Handbook Certificates The most important resource Ribbons and participation included in the School Competi- certificates for each regis- tion Kit. Includes 200 problems. tered student. You’ll also get access to electronic resources including 50 more handbook problems. The following are available to coaches at www.mathcounts.org/coaches. This section of the MATHCOUNTS website is restricted to coaches and you already should have received an email with login instructions. If you have not received this email, please contact us at [emailprotected] to confirm we have your correct email address. 2020 MATHCOUNTS School, MATHCOUNTS Practice Plans MATHCOUNTS Chapter + State Competitions for Team Meetings Problem of the Week Released each Monday Released by mid-April 2020 Pre-planned virtual mini-lessons, Each multi-step problem Each level includes all 4 test rounds each 45-60 minutes, that cover a relates to a timely event and the answer key particular math topic. You can use any or all of these resources to choose the students who will represent your school at the Chap- ter Competition. It is especially important though that your official competitors utilize the online 2020-2021 practice compe- titions so they are familiar with the online competition platform. The 2020-2021 MATHCOUNTS School Handbook will be COACHES: log in at your primary resource for the Competition Series this year. It is mathcounts.org/coaches designed to help your students prepare for each of the rounds of the test, plus build critical thinking and problem-solving skills. to get the problems, This section of the Guide for New Coaches will focus on how to answers, step-by-step solu- use this resource effectively for your team. tions and problem index for WHAT’S IN THE HANDBOOK? There is a lot included in the problems 201-250. School Handbook, and you can find a full table of contents on pg. 9 of this book, but below are the sections that you’ll use the most when coaching your students. • Handbook Problems: 250 math problems (200 in the book and 50 online) divided into Warm-Ups, Workouts and MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 5

Stretches. These problems increase in difficulty as the students progress through the book. (pg. 12) • Solutions to Handbook Problems: complete step-by-step explanations for how each problem can be solved. These detailed explanations are only available to registered coaches. (pg. 36) • Problem Index + Common Core State Standards Mapping: catalog of all handbook problems or- ganized by topic, difficulty rating and mapping to Common Core State Standards. (pg. 50) • Answers to Handbook Problems: key available to the general public. Your students can access this key, but not the full solutions to the problems. (pg. 52) There are 3 types of handbook problems to prepare students for each of the rounds of the competition. You’ll want to have your students practice all of these types of problems. Warm-Ups Workouts Stretches 14 Warm-Ups in handbook 8 Workouts in handbook 3 Stretches in handbook 10 questions per Warm-Up 10 questions per Workout Number of questions and use of calculators vary by Stretch no calculators used calculators used Warm-Ups prepare students Workouts prepare students Each Stretch covers a particular particularly for the Sprint particularly for the Target math topic that could be covered Round. Round. in any round. These help prepare students for all rounds. VS IS THERE A SCHEDULE I SHOULD FOLLOW FOR THE YEAR? On average coaches meet with their students for an hour once a week at the beginning of the year, and more often as the competitions approach. Practice sessions may be held virtually or in-person, before school, during lunch, after school, on weekends or at other times, coordinating with your school’s schedule and avoiding conflicts with other activities. Designing a schedule for your practices will help ensure you’re able to cover more problems and prepare your students for competitions. We’ve designed the School Handbook with this in mind. Below is a suggested schedule for the program year that mixes in Warm-Ups, Workouts and Stretches from the School Handbook, plus free practice competitions from last year. This schedule allows your students to tackle more difficult prob- lems as the Chapter Competition approaches. Mid-August – October 2020 November 2020 December 2020 September 2020 Warm-Ups 4 + 5 Warm-Ups 6 + 7 Warm-Ups 8 + 9 Warm-Ups 1, 2 + 3 Workout 3 Workout 4 Workout 5 Workouts 1 + 2 Mixture Stretch Statistics Stretch Pascal’s Triangle Stretch Practice Competition 1 Practice Competition 2 Practice Competition 3 January 2021 February 2021 Warm-Ups 10 +11 and Workout 6 2021 MATHCOUNTS Chapter Competition Past Competitions (2020 School + Chapter) 2021 MATHCOUNTS Chapter Invitational Practice Competition 4 Select chapter competitors (required by this time) FINISHED ALL THE PROBLEMS IN THE HARD-COPY HANDBOOK? FIND 50 MORE CHALLENGING PROBLEMS AT WWW.MATHCOUNTS.ORG/COACHES You’ll notice that in January you’ll need to select the 1-15 student(s) who will represent your school in the Chapter Competition. This must be done by January 15, 2021. You’ll identify your official competitors on the Art of Problem Solving (AoPS) Contest Platform. 6 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

It’s possible you and your students will meet more frequently than Check out the Interactive once a week and need additional resources. If that happens, don’t MATHCOUNTS Platform to get worry! You and your Mathletes can work together using the In- teractive MATHCOUNTS Platform, powered by NextThought. even more handbook This free online platform contains numerous MATHCOUNTS problems + past School Handbooks and past competitions, not to mention lots of competitions! features that make it easy for students to collaborate with each other and track their progress. You and your Mathletes can sign up for free at mathcounts.nextthought.com. And remember, just because you and your students will meet once a week doesn’t mean your students can only prepare for MATHCOUNTS one day per week. Many coaches assign “home- work” during the week so they can keep their students engaged in problem solving outside of team practices. Here’s one example of what a 2-week span of practices in the middle of the program year could look like. Monday Tuesday Wednesday Thursday Friday (Weekly Team Practice) -Students con- -Students continue to -Coach reviews solutions to -Coach emails -Students tinue to work work on Workout 4 Workout 4 math team to continue to individually on -Coach emails team -Coach gives Warm-Up 7 to assign Workout work indi- Workout 4, due to assign new home- students as timed practice and 5 as individ- vidually on Wednesday work problem(s), due then reviews solutions ual work, due Workout 5 Wednesday -Students discuss solutions to Wednesday homework problem(s) in groups -Students con- -Students continue to -Coach reviews solutions to -Coach emails -Students tinue to work work on Workout 5 Workout 5 math team to work to- individually on -Coach emails team -Coach gives Warm-Up 8 to assign Work- gether on Workout 5, due to assign new home- students as timed practice out 6 as group Workout 6 Wednesday work problem(s), due and then reviews solutions work, due using online Wednesday -Students discuss solutions to Wednesday Interactive homework problem(s) in groups Platform WHAT SHOULD MY TEAM PRACTICES LOOK LIKE? Obviously every school, coach and group of stu- dents is different, and after a few practices you’ll likely find out what works and what doesn’t for your students. Here are some suggestions from veteran coaches about what makes for a productive practice. • Encourage discussion of the problems so that students learn from each other • Encourage a variety of methods for solving problems • Have students write math problems for each other to solve • Practice working in groups so students can learn from one another • Practice oral presentations to reinforce understanding On the following page is a sample agenda for a 1-hour practice session. There are many ways you can struc- ture math team meetings and you will likely come up with an agenda that works better for you and your group. It also is probably a good idea to vary the structure of your meetings as the program year progresses. MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 7

MATHCOUNTS Team Practice Sample Agenda – 1 Hour Review Assigned Homework Problem(s) (10 minutes) • Have 1 student come to the board to show how s/he solved the problem(s). • Have students divide into groups of 4 to discuss the solution presented and other methods for solving. • Have 2 groups share an alternate solution. • Discuss as a group other strategies to solve the problem (and help if any groups answered incorrectly). Timed Practice with Warm-Up (15 minutes) • Have students put away all calculators and have one student Get more resources + pass out Warm-Ups (face-down). • Give students 12 minutes to complete as much of the Warm- activities to make team Up as they can. meetings fun at the coach section of the MATHCOUNTS • After 12 minutes is up, have students hold up pencils and stop working. Play Game to Review Warm-Up Answers (35 minutes) website! • Have students divide into 5 groups (size will depend on num- ber of students in meeting). • Choose a group at random to start and then rotate clockwise to give each group a turn to answer a question. When it is a group’s turn, ask the group one question from the Warm-Up. • Have the group members consult their completed Warm-Ups and work with each other for a maximum of 60 seconds to choose the group’s official answer. • Award 2 points for a correct answer on questions 1-3, 3 points for questions 4-7 and 5 points for ques- tions 8-10. The group gets 0 points if they answer incorrectly or do not answer in 60 seconds. • Have all students check their Warm-Up answers as they play. • Go over solutions to select Warm-Up problems that many students on the team got wrong. OK I’M READY TO START. HOW DO I GET STUDENTS TO JOIN? Here are some tips given to us from successful competition coaches and club leaders for getting students involved in the program at the begin- ning of the year. • Ask Mathletes who have participated in the past to talk to other students about participating. • Ask teachers, parent volunteers and counselors to help you recruit. • Reach parents through school newsletters, PTA meetings or Back-to-School-Night presentations. • Advertise around your school by: 1. posting intriguing math questions (specific to your school) in your school and referring students to the first meeting for answers. 2. designing a bulletin board or display case with your MATHCOUNTS poster (included in your School Competition Kit) and/or photos and awards from past years. 3. attending meetings of other extracurricular clubs (such as honor society) so you can invite their members to participate. 4. adding information about the MATHCOUNTS team to the website for your class or school. 5. making a presentation at the first pep rally or student assembly. Good luck in the competition! If you have any questions during the year, please contact the MATHCOUNTS national office at [emailprotected]. COACH RESOURCES: WWW.MATHCOUNTS.ORG/COACHES 8 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

2020-2021 Handbook Materials Thank you for being a coach in the MATHCOUNTS Competition Series this year! We hope participating in the program is meaningful and enriching for you and your Mathletes. Don’t forget to log in at www.mathcounts.org/coaches for additional resources! WHAT’S IN THIS YEAR’S HANDBOOK Highlighted Resources ............................................................................................... 10 Best Materials + Tools for Coaches and Mathletes! Critical 2020–2021 Dates ........................................................................................ 11 This Year’s Handbook Problems ....................................................................... 12 200 Math Problems to Boost Problem-Solving Skills (+50 more online!) Competition Coach Toolkit ................................................................................. 33 Vocabulary, Formulas + Tips Organized by Math Topic Solutions to Handbook Problems.................................................................... 36 Step-by-Step Solution Explanations for Coaches Problem Index + Common Core State Standards Mapping ......................... 50 All 200 Problems Categorized + Mapped to the CCSS Answers to Handbook Problems ..................................................................... 52 Available to the General Public...Including Students COACHES: FIND PROBLEMS, ANSWERS, SOLUTIONS + PROBLEM INDEX FOR#201-250 ONLINE AT WWW.MATHCOUNTS.ORG/COACHES MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 9

Highlighted Resources ACCESS MORE RESOURCES AT WWW.MATHCOUNTS.ORG/COACHES MATHCOUNTS OPLET The Online Problem Library & Extraction Tool (OPLET) is a database of over 13,000 problems and over 5,000 step-by-step solutions. Create personalized quizzes, flash cards, worksheets and more! Save $25 when you buy your subscription by Oct. 1, 2020 www.mathcounts.org/myoplet PRACTICE COMPETITIONS FOR MATHCOUNTS, VOL. 1 & 2 Practice books written by repeat national-level coach Josh Frost. Each volume includes 4 complete mock-competitions plus solutions. www.mathcounts.org/store MOST CHALLENGING MATHCOUNTS PROBLEMS SOLVED, VOL. 2 This book contains the Sprint Round and Target Round problems from the 2011 – 2019 MATHCOUNTS National Competitions plus never-before-available, step-by-step solutions! COMING SEPTEMBER 2020! 10 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Critical 2020-2021 Dates Jun. 16 – Submit your school’s registration to participate in the Competition Series. The fastest way for Jan. 15 schools to register is online at www.mathcounts.org/compreg. Schools also can download the MATHCOUNTS Competition Series Registration form and mail or email it with payment to: Oct. 1 MATHCOUNTS Foundation – Competition Series Registrations (postmark) 1420 King Street, Alexandria, VA 22314 Email: [emailprotected] Oct. 15 Parents who have confirmed their student’s school is not participating in the program can register Nov. 15 their child as a non-school competitor (NSC) this year. NSCs must register online and pay with a Dec. 1 credit card at www.mathcounts.org/nscreg. (postmark) Registered schools and NSCs will receive this year’s School Competition Kit, which includes a hard copy of the 2020-2021 MATHCOUNTS School Handbook. Kits are shipped on an ongoing Dec. 15 basis between mid-August and January 31. Early Jan. Questions? Email the MATHCOUNTS national office at [emailprotected]. Jan. 15 . (postmark) Early Bird Registration Deadline ($30/student for schools and $60/NSC) Jan. 22- After October 1, registration increases to $35/student for schools and $65/NSC. 23 Feb. 5-6 . Feb. 25 Mar. 25 Practice Competition 1 Released, available for registered schools and NSCs at the Art of May 8-11 Problem Solving (AoPS) Contest Platform through January 31, 2021. . Practice Competition 2 Released, available for registered schools and NSCs at the AoPS Contest Platform through January 31, 2021. . Regular Registration Deadline ($35/student for schools and $65/NSC) After December 1, registration will cost $40/student for schools and $70/NSC. . Practice Competition 3 Released, available for registered schools and NSCs at the AoPS Contest Platform through January 31, 2021. . Coaches should identify official chapter competitors on the AoPS Contest Platform at this time to ensure their students can practice using the competition platform. Identify official competitors by January 15 to ensure their participation in Practice Competition 4. . Final Day to Register, Pay and Add Students ($40/student for schools and $70/NSC) MATHCOUNTS cannot guarantee participation for schools with outstanding invoices after Janu- ary 15. Schools cannot add students to their registration after January 15. . Practice Competition 4, available from 1:00pm ET on January 22 through 1:00pm ET on Jan- uary 23 on the AoPS Contest Platform. Recommended for chapter competitors to use as a run- through for the Chapter Competition. . 2021 Chapter Competition, available from 1:00pm ET on February 5 through 1:00pm ET on February 6 on the AoPS Contest Platform. . 2021 Chapter Invitational at 7:00pm ET on February 25 on the AoPS Contest Platform. All competitors must take this competition at the same time. . 2021 State Competition at 7:00pm ET on March 25 on the AoPS Contest Platform. All com- petitors must take this competition at the same time. . 2021 Raytheon Technologies MATHCOUNTS National Competition in Washington, DC MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 11

Mixture Stretch 1. _____t_ea_s_p_oo_n_s Ming’s recipe for sweet tea calls for 4 teaspoons of sugar. If Ming wants to make the tea 25% less sweet, how much less sugar should he use? 2. ___________ Carla is mixing cherry, grape and lime candies in a bowl. Since her favorite flavor is cherry, s41heowf tahnetsca25ndoiefsthtoe candies to be cherry. Since her least favorite flavor is lime, she wants be lime. What fraction of the candies will be grape? Express your answer as a common fraction. 3. ___________ A paving company makes concrete by adding water to a mix that is 1 part cement, 3 parts sand and 3 parts aggregate (stone). What fraction of this mix is aggregate? Express your answer as a common fraction. 4. _______o_u_nc_e_s A beef stew recipe calls for 12 ounces of beef, 4 ounces of carrots, 7 ounces of potatoes, 4 ounces of peas and 5 ounces of beef stock. Given that there are 16ounces in a pound, how many ounces of potatoes are needed to make 4pounds of this stew? 5. __________%_ Jin adds 1 gallon of a water-and-bleach mixture that is 4% bleach to 2 gallons of a water-and-bleach mixture that is 10% bleach. What percent of the final mixture is bleach? 6. _$__________ CASHEW Cashews cost $2.36 per pound, almonds cost $1.48 per pound and peanuts cost ALMPOENADNUT $0.98 per pound. To make a 20% profit, how much should Myrna charge per pound for a mixture that is 1 part cashews, 1 part almonds and 2 parts peanuts? 1 lb MIX 7. _______g_a_llo_n_s Manny’s cleaning supply store receives a mixture of 80% detergent and 20% water in 15-gallon buckets. Manny would like a mixture of 60% detergent and 40%water in 5-gallon buckets. To make this, he combines some 80/20 mixture with some pure water in each 5-gallon bucket. How many gallons of pure water does Manny add to each 5-gallon bucket? Express your answer as a decimal to the nearest hundredth. 8. _______b_uc_k_et_s Based on the information in problem 7, how many 5-gallon buckets of 60/40 solution can Manny make from one 15-gallon bucket of 80/20 solution? 9. ________q_ua_rt_s Dara is mixing her own paint color, using 3 parts green paint to 2 parts blue to 1 part white. Given that there are 4 quarts in a gallon, if she needs 3gallons of her paint, how many quarts of white paint should she buy? 10. _______g_/c_m_3 To make a sand sculpture, Arthur used 2 cm3 of red sand with a density of 4 g/cm3, 7cm3 of yellow sand with a density of 5 g/cm3 and 5 cm3 of brown sand with a density of 6 g/cm3. What is the average density of this sculpture in grams per cubic centimeter? Express your answer as a decimal to the nearest tenth. 12 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Statistics Stretch 11. __________ A class of 28 students had a mean score of 72 on a math test. After the teacher realized that one of the questions had an alternative correct answer, he gave 4 points each to the 7students who had given the alternative answer. What is the new mean test score? 12. _______p_ag_e_s stem leaf The stem-and-leaf plot shows the number of pages in each book that Kalem read last summer. How many pages did 1 27 52 82 Kalem read last summer? 2 25 57 63 82 97 3 37 51 68 75 1|27=127 pages 13. __________ Based on the information in problem 12, what portion of the pages that Kalem read were in books having more than 275 pages? Express your answer as a common fraction. 14. _________%_ A cafeteria offers apples, oranges and bananas with lunch. A student may take at most one of each fruit. Of the 61 students who got fruit with lunch, 5students got only an apple and 7 got only an orange. Of the 16 students who got an apple and an orange, the 17 who got an orange and a banana, and the 20 who got an apple and a banana, 6 got all three fruits. What portion of the fruit taken by the 61 students were bananas? Express your answer to the nearest whole percent. 15. __________ Chess Club Membership The table shows the grade and skill level of members of the chess club. If half of the members are eighth graders and 6th 7th 8th one-third of the beginners are seventh-graders, what fraction Beginners 1 ? 5 of chess club members are advanced chess players? Express Advanced 2 ? 6 your answer as a common fraction. 16. __________ If A, M and R represent the arithmetic mean, median and range of the set {13, 16, 18, 23, 25, 28, 30, 31}, what is the value of A + M − R? Cat Owner Survey Number of People 10 17. ________c_at_s This graph shows results of a survey of 25 cat owners. What 5 is the mean number of cats per person surveyed? Express your answer as a decimal to the nearest hundredth. 12345 Number of Cats 18. __________ A total of 44 Mathletes competed in a MATHCOUNTS competition. The mean score for all the competitors was 28. The mean score for all competitors except the 16 highest scorers was 20. What was the mean score for the 16 highest scorers? 19. _________%_ Bryce orders 6 bats, 60 baseballs and 8 gloves. If each bat costs $29.95, a pack of 12baseballs costs $39.95 and a glove costs $69.95, what portion of the total cost of this order is for the gloves? Express your answer to the nearest whole percent. 20. _________%_ The graph shows the price for a popular running shoe over Running Shoe Price five months. What is the absolute difference of the percent $90 change in price from February to April and the percent $80 change in price from April to June? Express your answer to $70 the nearest tenth of a percent. $60 Feb Mar Apr May Jun MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 13

Pascal’s Triangle Stretch Pascal’s triangle is a famous triangular array made up of the binomial coefficients. The rows are numbered 0, 1, 2, 3, …, and in row n there 1 Row 0 are n + 1 entries, numbered 0, 1, …, n. In each row, both the first 1 1 Row 1 entry (entry 0) and the last entryare 1. Each other entry is the sum 1 2 1 Row 2 of the twoentries above it (one to the left, the other to the right). 1 3 3 1 Row 3 Here we have a Pascal’s triangle with 16 rows, showing the 1 4 6 4 1 Row 4 entries for rows 0 through 11. Notice that the 10 in row5 is 1 5 10 10 5 1 Row 5 the sum of the 6 and 4 in row 4, directly above it. 1 6 15 20 15 6 1 Row 6 The binomial theorem says that when the expression 1 7 21 35 35 21 7 1 Row 7 ( ((x + y)n is expanded and like terms are combined,n 1 8 28 56 70 56 28 8 1 Row 8 the coefficient of xkyn − k is k , meaning that the 1 9 36 84 126 126 84 36 9 1 Row 9 coefficients in this expansion can be read directly 1 10 45 120 210 252 210 120 45 10 1 Row 10 from row n in Pascal’s triangle. For example, 1 11 55 165 330 462 462 330 165 55 11 1 Row 11 consider (x + y)4. Expanding this binomial, we get x4 + 4x3y + 6x2y2 + 4xy3 + y4. In this Row 12 case, n=4. Referring to Pascal’s triangle, Row 13 we see that row 4 does indeed give us Row 14 the coefficients of the terms in this Row 15 expansion: 1, 4, 6, 4 and 1. nnkum=bneCr ko=f wka!(ynsn−!tok)c!h,othoesenuthmrebeeroof fthceomm,boinr at53ion, s ( (In ( (of general, the kth entry in row n is the binomial coefficient n objects taken k at a time. Given five objects, to find the we locate entry 3 of row 5 in Pascal’s triangle and see that there are 10 ways. Below are a few interesting properties of Pascal’s triangle. Diagonal Patterns 1 Row Sums 1 1= 1 20 11 11 1+1= 2 21 121 121 1+2+1= 4 22 1331 1331 1+3+3+1= 8 23 14641 14641 1 + 4 + 6 + 4 + 1 = 16 24 1 5 10 10 5 1 1 5 10 10 5 1 1 + 5 + 10 + 10 + 5 + 1 = 32 25 1 6 15 20 15 6 1 The sum of the entries in the nth row of Pascal’s triangle is equal to 2n. 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 Hockey Stick 1 The hockey stick identity: Start at any 1 1 of the 1s on the left or right side The gray diagonal contains the counting 1 2 1 of Pascal’s triangle. Sum entries numbers: 1, 2, 3, 4, 5, …. 1 3 3 1 diagonally in a straight line, and 1 4 6 4 1 stop at any time. The next entry The black diagonal contains the 1 5 10 10 5 1 down diagonally in the opposite triangular numbers: 1, 3, 6, 10,15, …. In fact, entry2 in row(n+1) is the nth direction will equal that sum. triangular number. 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 14 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Solve the following problems, using what you’ve learned about Pascal’s triangle. It may be helpful to fill in some of the missing entries in the Pascal’s triangle on the previous page. 21. __________ What is the greatest entry in row 15 of Pascal’s triangle? 22. ______e_n_tr_ie_s How many of the entries in row 14 of Pascal’s triangle are even? 23. __________ What is the sum of the entries in row 12 of Pascal’s triangle? 24. ______c_ho_ic_e_s A sports team of eight players must choose three starting players. How many different choices of three starters are there if the order in which they are chosen does not matter? 25. __________ What is the sum of the first 10 triangular numbers? 26. __________ When the expression (x + 2)8 is expanded and like terms are combined, what is the coefficient of x3? 27. __________ When the expression (2x + y)4 is expanded and like terms are combined, what is the sum of the coefficients? 28. __________ When the expression (x + 1)2022 is expanded and like terms are combined, the term with the greatest coefficient can be expressed as axb. What is the value of b? 29. __________ A fair coin is flipped four times. What is the probability that it lands heads up at least as many times as it lands tails up? Express your answer as a common fraction. 30. _______t_im_e_s Only the number 1 appears in Pascal’s triangle more times than the number 3003 appears. How many times does 3003 appear in Pascal’s triangle? MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 15

Warm-Up 1 31. __________ What is the result when one hundred twenty-eight thousand is subtracted from onemillion? 32. ________u_n_it How many unit cubes are required to create a larger cube with edge cubes length 3 units? 33. __________ What is the remainder when the sum of the smallest and second-smallest prime numbers is divided by the third-smallest prime number? X Y 34. ______de_g_re_e_s In parallelogram WXYZ, shown here, the measure of angle W is 80degrees. What is the degree measure of angle X? W 80° Z 35. __________ What is the value of 6 ÷ 2 × 3 + 8 ÷ 4 × 2? 36. __________ If Yasuko randomly selects a single-digit positive integer, what is the probability that it is not prime? Express your answer as a common fraction. 37. __________ What is the value of 0.001 × 1 × 105? 10 38. __________ What is the value of 3(4x + 5y) − 2(7x − 3y) when x = −2 and y=3? 39. __________ If 4 = a , what is the value of a? 18 27 40. ________cu_p_s Daif*cku is a snack made from glutinous rice flour and sweetened red bean FF paste. A recipe for 24 daif*cku requires 3 cups of sugar. How many cups LL of sugar are needed to make 64 daif*cku? OO UU Sugar RR 16 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Warm-Up 2 41. ____:____p_.m_. Louisa leaves her house at 12:17 p.m., walks to the library, which takes 14 minutes, and remains there for 3 hours 27 minutes. If she walks home at the same speed as she walked to the library, at what time will she return home? 42. __________ What is the value of 7 − (3 − 4) + 11? 43. __________ What is the least positive integer that is divisible by 4, 6 and 10? 44. _______y_a_rd_s How many yards are in the perimeter of a square that measures 99inches on each side? 45. ______g_a_llo_n_s When the Schwartzes left home, their rgeaasdg41aufguell.rIefathde7i8r full. When they reached their destination, their gauge gas tank holds 16gallons, how many gallons of gas did they use on their trip? 46. __________ What is the eighth term of the sequence that begins with 1, 3, 7, 13, 21, …? 47. __________ What is the value of x that satisfies the equation 5(x + 2) − 3(x − 8) = 16? 48. ______m_in_u_te_s Andrew mowed one-half of a lawn, and Ben mowed one-third of the same lawn, each at a constant rate. If Andrew, continuing at the same rate, finished mowing the rest of the lawn in 12 minutes, how many minutes would it have taken him to mow the entire lawn by himself? 49. __________ If 125% of n is 30, what is 25% of n? 50. _______o_u_tfi_ts Don has four short-sleeved shirts, one each in black, white, red and gray, and two long-sleeved shirts, one each in red and gray. Don has three pairs of pants, one each in gray, black and tan. If Don chooses to wear only one black item or no black items at any one time, how many different outfits consisting of a shirt and a pair of pants can he make? MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 17

Warm-Up 3 51. _______t_im_e_s The face of a clock has the numbers 1 through 12 painted on it. How many times is the digit 1 painted? 52. __________ What is the value of 5 + (–6) + 5 – (–6) + 5 – 6 + (–5 + 6)? y x 53. __________ What is the sum of the coordinates of point D on the coordinate grid 6 shown? 4 2D −2 2 4 6 −2 ( – ) ( – )54.__________ What is the value of 1 1 21 1 2 23 ? Express your answer as a common fraction. 55. __________ What is the value of 0.123 + 1.032 + 2.301 + 3.210? Express your answer as a decimal to the nearest thousandth. 56. ______p_r_im_e_s Let p(n) be the number of primes less than n. What is p(50)? 57. _______c_u_p_s Maria wants to make a casserole to serve 12 people. She plans to use a 1 recipe that calls for 1 4 cups of flour to serve 5 people. How many cups of flour will Maria need for her casserole? x 58. __________ Let x and y represent the LCM and GCF, respectively, of 24 and 40. What is the value of y ? ℓ 59. ______de_g_re_e_s The figure shows parallel lines m and n with transversal ‫ٴ‬. Based m 2x + 6 on the degree angle measures shown, what is the value of x? x n 60. __________ What is the greatest two-digit prime number that is one greater than a perfect square? 18 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Warm-Up 4 61. _______g_ra_m_s Each of 6 Mathletes ate a cheeseburger that contained 63 calories of fat. If there are 9calories per gram of fat, how many total grams of fat did the Mathletes eat? 62. __________ What is the value of the 40th positive odd integer? 63. ________un_it_s What is the distance between C(2, 1) and D(5, 5)? 64. __________ 12 = 1 Based on the pattern shown, what is the sum of the digits when 112 = 121 11,1112 is calculated? 1112 = 12,321 11112 = 1,234,321 65. __________ What is the value of 1 + 2 × 3 – 4 + 5 × 6 – 7 + 8 × 9? 66. _________%_ Last school year, the Math Club had 20 members. This school year, there are 28 members. By what percent did the membership of the Math Club increase? 67. _________in_2 What is the area of a rectangle that has width 2 3 inches and length 3 2 inches? Express your answer as a mixed number. 4 5 68. __________ When the grid shown is filled in correctly, each of the numbers 1 through 4 743 will appear exactly once in each row and column. The small number in one 63 corner of a heavily outlined rectangle is the sum of the numbers that belong in that rectangle. What number must be in the shaded cell of the grid? 7 46 1+ 2 2− 69. __________ What is the value of 3 ? Express your answer as a common fraction. 3 4 70. __________ Let x 5y = (x + 8)2 . What is the value of 2 5 (2 5 4)? y MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 19

Warm-Up 5 71. _______p_oi_nt_s In the National Sports League (NSL), a team earns 3 points for a regulation win, 2 points for an overtime win and 1 point for an overtime loss. How many total points did an NSL team with 29 regulation wins, 10 overtime wins and 4 overtime losses earn? 72. __________ What common fraction is equivalent to 48.55 − 47.37? 73. __________ If s equals the square root of the reciprocal of 1.21, what is the value of s? Express your answer as a common fraction. 74. _______p_oi_nt_s A group of 4 students took a math test. The mean of the numbers of points scored by the students is 95 points out of a possible 100 points. What is the minimum possible number of points scored by any student? A MP B NO 75. __________ In square ABCD, shown here, segments MN and OP trisect sides AB R and DC, and segments QR and ST trisect sides AD and BC. What Q is the ratio of the combined area of the shaded regions to the area of T S square ABCD? Express your answer as a common fraction. D C 76. __________ All the students at the playground are wearing either long pants or shorts, and some are wearing a hat. Here is a table where Mossi recorded how many students are wearing various items. What is the probability that a randomly selected student on Hat No Hat this playground is wearing shorts but no hat? Express your answer as a common fraction. Pants Shorts 77. __________ What is the result when −8 × (−4) − (−8) is divided by the sum −6 + (−4)? 78. _$_________ Simon and Theo paid a total of $15 for lunch. Simon paid 2 the amount that Theo paid. How many more dollars than Simon did Theo pay? 3 79. _________f_t3 Molly is folding a right triangular prism out of a piece of poster board cut into A the shape shown. If the distance between point A and point B is 24 inches, B what will the volume of the prism be in cubic feet? 80. __________ What is the value of 202,020,202,020 × 2021 − 202,120,212,021 × 2020? 20 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Warm-Up 6 81. __________ If A = 23 − 32 + 4(5 + 1) and B = 72 − 2(3 + 1)2, what is the value of A − B? EF A 82. __________ What is the ratio of the area of square ABCD to that of square BDEF B inthe figure shown? Express your answer as a common fraction. D C 83. __________ What is the median of the first ten prime numbers? 84. __________ Let the value of a word equal the sum of the values of its letters. Suppose MATH has value M + A + T + H = 85, and suppose M = 8, A = 3H, T = 73 and H = 1. If S = 3M − 15 and N = 2H+ S, what is the value of SAMANTHA? 85. __________ What is the remainder when the sum 103 + 42 is divided by 8? 86. __________ Hugo has a solid like the one shown, composed of 32 unit cubes. He paints each face of this solid red and then separates the solid into the 32 individual unit cubes. What is the probability that a randomly selected unit cube has exactly one painted face? Express your answer as a common fraction. 87. __________ If x equals twice its reciprocal, what is the value of x4? 88. _______ro_o_m_s A robotic vacuum cleaner vacuums one hotel room in 3 hour. At this rate, how many rooms of the same size will it vacuum in 3 hours? 10 89. __________ Nya starts with an integer N and repeatedly subtracts 6. Mya starts with an integer M and repeatedly adds 8. When Nya and Mya have each performed their respective repeated operation 13 times, both have a resulting value of 25. What is the value of N + M? ( (( (( ( ( (90.__________ 1 1 1 ‫ڮ‬ 1 What is the value of the product 1 + 2 1+ 3 1+ 4 1+ 19 ? MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 21

Warm-Up 7 91. _____s_tu_d_e_nt_s This school term, 25% of the students in Ms. Norton’s class earned a final grade of A. If 7students earned an A this term, how many students are in Ms. Norton’s class? 92. ________c_m_2 The figure shows an isosceles triangle inscribed in a semicircle of radius 10cm. What is the area of the triangle? 10 cm 93. __________ What is the integer value of 1.4×105 ? 7×102 Maeve’s Tiles 94. ________t_ile_s dbeyspiganintbinygpa95intoinf ghis31sqouf ahreer Maeve creates a square tiles. Liam creates a design tiles. What is the Liam’s Tiles combined number of painted tiles in Maeve’s and Liam’s designs? 95. __________ If a = 2 and a + b = 100, what is the value of b? b 3 96. __________ The mean, median and unique mode of a list of five positive integers are all equal to 5. What is the greatest possible value of an integer in this list? 97. __________ Leah puts a toy car weighing 6 ounces on the left side of a balance. Then, reaching into a bag that contains four weights, measuring 1, 2, 4 and 5ounces, she randomly removes two weights, without replacement. If she places the two weights on the right side of the balance, what is the probability that the balance levels? Express your answer as a common fraction. 98. __________ What is the value of 74 − 34 ? 72 + 32 99. _____t_ri_an_g_le_s How many triangles of any size are in the figure shown? 100.____a_r_ra_ng_e_- Sara and Ben make a playlist for a road trip. Each chooses 5 songs for the playlist, and ments they order the songs so that no two consecutive songs were added to the list by the same person. How many such song arrangements are possible for their playlist, assuming no song is repeated? 22 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Warm-Up 8 101.__________ What number is one-third of two-fifths of 90? 1 102._______u_ni_ts_3 What is the volume of the figure shown, in which all adjacent edges are perpendicular? 5 2 34 103.__________ When a number n is divided by −6 and the quotient is increased by 6, the result is 3. What is the value of n? 104.__________ What common fraction is equivalent to 1 + 1 ? 1+ 1 1 1+ 3 105.__________ What is the absolute difference between the slope of the line passing xy through the points given in this table and the slope of the line given by2x − y = 4? Express your answer as a common fraction. 12 47 7 12 106.__________ What is the value of 25 × 4–2? 107._______y_e_ar_s Armin’s 13th birthday was on Saturday, July 4, 2020. How old will Armin be when his birthday next falls on a Saturday? old 108.__________ What is the value of a in the geometric sequence −1, 3, −9, a, −81? 109.__________ Suppose that N is an integer such that 3N is a factor of 10!. What is the greatest possible value of N? AD 110.________un_it_s In the figure shown, CD = 6 units, m²CAD = 30°, m²ACE = 45° and C m²ABC = 60°. What is the length of segment EB? Express your answer E in simplest radical form. B MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 23

111.__________ Warm-Up 9 A bag contains 12 hair bows: 5 red, 4 white and 3 blue. Jo Jo reaches into the bag and randomly pulls out two bows without replacement. What is the probability that those two bows are the same color? Express your answer as a common fraction. ( (112.__________If 4 5 x− 1 = 3, what is the value of x? 8 2 113.__________ A fair coin is flipped 5 times. What is the probability that no two consecutive flips have the same result? Express your answer as a common fraction. 114.__________ A right triangle has side lengths x, 3x and 10, as shown. What is x 3x the value of x? Express your answer in simplest radical form. 10 115.__________ What is the value of 412 × 1257? Express your answer in scientific notation. 116._________cm_ Rachelle draws a rectangle of perimeter 46 cm and area 90 cm2. Evan draws a rectangle with twice the perimeter and half the area of Rachelle’s rectangle. What is the smaller dimension of Evan’s rectangle? 117._______c_o_in_s Tonya found $2.25 in nickels and quarters in her sofa cushions. If the number of nickels Tonya found is five more than three times the number of quarters she found, what is the total number of coins Tonya found? 118.__________ A line passes through the points (−7, 1), (5, 7) and (0,b). What is the value of b? Express your answer as a common fraction. 119.__________ Every positive integer can be expressed in the form 6n + k, where 0 ൑ k ൑ 5. If 1841 is expressed in this form, what is the value of n + k? 120._________in_3 A lamp’s base and shade are both cylindrical as shown. The shade has Shade Base circumference 18π inches, which is three times that of the lamp’s base. If the lamp’s base is made of solid brass and has height 9inches, what is 9 in the volume of brass in the lamp’s base? Express your answer in terms of π. 24 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Warm-Up 10 A FD 8 E 121.________c_m_2 Square ABCD, shown here, has side length 8 cm. If E and F are the B C midpoints of sides CD and AD, respectively, what is the area of shaded trapezoid ACEF? 122.__________ If f(x) = 2x3 – 5x2 + 9x + 4, what is the value of f(–2)? 123.__________ Three numbers are selected at random without replacement from the set {2, 3, 5, 7, 11,13}. What is the probability that the sum of the three numbers will be a multiple of 3? Express your answer as a common fraction. 124.__________ In the square shown, stripes run parallel to the sides and divide the top and right sides of the square into congruent segments. What fraction of the figure is shaded? Express your answer as a common fraction. ( )125.__________ 1 ⎡ 2 2 ⎤2 ⎣⎢ 3 4⎥ What is the value of ÷ ? ⎦ 126.________c_m_2 The right triangular prism shown has bases that are isosceles triangles. If each triangular base has congruent sides of length 5cm and a third side of length 6 cm, and the prism has height 3cm, what is the surface area ofthis prism? 127.__________ What is the smallest five-digit number that has exactly one 4 and exactly one 6 and is a multiple of 9? 128.__________ AB C In the circular spinner shown, sections A and B are congruent, each ED with a 90-degree central angle, and sections C, D and E are all congruent. When Allie spins this spinner, what is the probability that it lands on section A or D? Express your answer as a common fraction. 129.__________ Two numbers have a geometric mean of 4 and an arithmetic mean of 5. What is the larger of the two numbers? T 130._______p_a_th_s In the figure shown, how many paths from B to T are there that move up and right along the line segments? B MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 25

Warm-Up 11 131.__________ If f(x) = x2 and g(x) = 2x − 1, what is the value of f(5) − g(8)? 132.__________ A nugget number is a positive integer that can be obtained by adding together any combination of the numbers 6, 9 and 20. For example, 75 is a nugget number because 20+ 20 + 20 + 9 + 6 = 75, whereas, 34 is not a nugget number. What is the largest nugget number less than 200? 3 133.__________ What is the value of 81 4 ? 134._____a_rr_a_ng_e_- How many unique arrangements are there of all six letters of SQUARE? ments 135._______w_a_y_s In how many ways can all the numbers 1, 2, 3, 4, 5, 6 and 7 be separated into two groups, so that the sum of the numbers in both groups is the same? 136.__________ If x + 1 = –2, what is the value of x4 + 1 ? x x4 137.______b_ur_g_er_s Bodacious Burgers sells burgers for $2.50 each and french fries for $0.99 per container. Four friends ate at Bodacious Burgers. The bill for the meal was $17.97 before tax. How many burgers were ordered? 138._________cm_ A right circular cone with base radius 4 cm has surface area 56π cm2. What is h the height of the cone? Express your answer in simplest radical form. 4 139.__________ What is the least positive two-digit integer that leaves a remainder of 3 when divided by each of the numbers 4, 5 and 6? 140.________un_it_s A triangle has sides of lengths 18, 24 and 30 units. What is the length of the shortest altitude of this triangle? Express your answer as a common fraction. 26 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Workout 1 141.__________ What is the sum of the integers less than 100 that have both 6 and 9 as divisors? 142.__________ Devon wrote a program that takes a positive integer A as an input and performs a series of operations, each time assigning the result to a new variable, as shown. If the output of the program is 144, what was the value of the input A? INPUT C E G A B2 5 D 10% of F 6 BD F OUTPUT 1 A 6C E +100 12G 5 143.__________ What is the arithmetic mean of the numbers 3, 66 and 999? 144._______h_o_ur_s Grace notices interstate highway markers placed at the end of each mile and numbered consecutively: 1, 2, 3, 4, …. If an accurate speedometer says Grace is traveling 72 mi/h, how many hours will it take her to travel from mile marker 7 to mile marker 29? Express your answer as a decimal to the nearest hundredth. 145._________%_ Cliff’s piano has 52 white keys and 36 black keys. What percent of the keys on his piano are white? Express your answer to the nearest whole percent. 146._________m_L On average, in men, 39 mL per kilogram of body weight is blood plasma. In women, 40mL per kilogram of body weight is blood plasma. Rob weighs 80 kg, and Kristen weighs 60kg. How many more milliliters of blood plasma does Rob have than Kristen? 147._______t_im_e_s A CD spins 5 times per second. How many times will the CD spin while playing a song that is 3 minutes 43 seconds long? 148.__________ What is the greatest possible product of two positive integers whose sum is 34? Summer Break Reading 149._______b_oo_k_s The line plot shows the number of books that each student in Ms. Coleman’s homeroom reported reading over summer break. What was the mean number of books read by a student in this homeroom? Express your answer as a decimal to the nearest tenth. 0 1 2 3 4 5 6 7 8 Number of Books Read 150._____d_e_g_re_e_s Angles A and B are complementary, with m²A = 5x – 6 degrees and m²B = 3x degrees. What is the degree measure of angle A? MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 27

Workout 2 151.________f_ee_t If 1 inch is equivalent to 2.54 cm, how many feet are in 50 cm? Express your answer as a decimal to the nearest hundredth. 152.______in_te_g_er_s How many two-digit positive integers are there with the property that the sum of the integer’s digits equals the product of those digits? 153.__$________ Alistair has 40 coins in his pocket, including at least one penny, one nickel, one dime and one quarter. If he has no other type of coin in his pocket, what is the greatest possible total value of the coins in Alistair’s pocket? 154._________%_ If Sanjay runs 80% as far as Jerome in the eighth-grade pickle-rolling race, what percent of Sanjay’s distance does Jerome run? 155.__________ What is the least common multiple of the first five positive cubes? 156._______u_n_its_2 What is the area of the triangle bounded by the line 3x + 2y = 12 and the x- and y-axes? 157._______h_o_ur_s The table shows the hours that Gene, Doug and Pat worked canning corn one Saturday in July. What is the combined number of hours they worked canning corn? Express your answer as a decimal to the nearest hundredth. 1st Shift 2nd Shift GENE Start End Start End DOUG 7:00 a.m. 10:15 a.m. 10:30 a.m. 12:30 p.m. 7:30 a.m. 10:15 a.m. 10:30 a.m. 12:30 p.m. PAT 7:15 a.m. 10:15 a.m. 10:30 a.m. 11:45 a.m. 158.__________ What is the median of the integers between 1 and 1000 that are divisible by 28? 159.________un_it_s The figure shows a hexagon in which adjacent sides are perpendicularto 5 12.5 each other. The hexagon with the given side lengths has an area of a 15.8 160units2. What is the value of aԝ? Express your answer as a decimal to the nearest tenth. 160.______p_r_im_e_s Let an optimus prime be a prime number whose digit sum is also a prime number. How many of the first 10 primes are optimus primes? 28 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Workout 3 { }161.__________ 7 1 1 What is the median of the set 3 , 2, 1.5, 4 ? Express your answer as a decimal to the nearest hundredth. 162._______y_e_ar_s Marco’s age is between 20 and 60. The square of his age is a four-digit number, and the old sum of the digits of his age is 8. How old is Marco if his age has two different digits and is not prime? 163.________f_ee_t The flagpole shown is perpendicular to the ground and casts a shadow of 60° 8feet. The angle of elevation to the top of the pole from the end of the shadow is 60 degrees. How tall is the pole? Express your answer as a 8 ft decimal to the nearest tenth. 164._______b_lu_e_- The number of blueberries left in a bowl is reduced by half every 2 hours. Sam filled the berries bowl with blueberries at 9:00 a.m. When he checked the bowl at 7:00 p.m., there were 5blueberries left. How many blueberries did Sam originally place in the bowl at 9:00a.m.? C 165._____is_o_sc_e_le_s α In isosceles triangle ABC, shown here, AB = AC. For the angles θ indicated, ˙ = 36°, ˠ = 72° and ˭ = 108°. How many isosceles triangles triangles of any size are in this figure? φ Aα θφ B 166.______in_te_g_er_s How many integers between 1,000,000 and 2,000,000 are divisible by 99? 167.______g_a_llo_n_s An acre-foot is defined as the volume of a rectangular prismwitha base areaof1acre and a depth of 1 foot, as shown. Given that 1 acre is defined as the 1 ft areaofa 66-foot×660-foot rectangle and that 1 gallon equals 231in3,how manygallons are in 1 acre-foot of water? Express youranswer asan integer to the nearestthousand. V = 1 acre-foot 168._________%_ What percent of positive integers less than or equal to 100 are divisible by 3? 169._____s_tu_d_e_nt_s Ali’s middle school has a total of 300 students on Team A and Team B. After 30 students are moved from Team A to Team B, there are twice as many students on Team A as there are on Team B. How many students were originally on Team B? 170.________c_at_s The mean number of cats living in each of the 50 apartments in a particular apartment building is 0.44cats. A total of 32 apartments in the building are cat-free. What is the mean number of cats in the apartments that have at least one cat? Express your answer to the nearest tenth. MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 29

Workout 4 171.__________ What common fraction t satisfies the equation t 1 = 4 ? 3t + 5 172._______w_a_y_s Mr. Scott has five algebra books and four geometry books. He wants to arrange them all on a single shelf. If Mr. Scott keeps all of the algebra books together and all of the geometry books together, how many ways can he arrange these books on his shelf? 173.__________ If A + B = C + 1, B + C = D – 1, C + D = E + 1, D + E = F – 1, E + F = G + 1, F+G= A – 1 and G + A = B + 1, what is the value of A + B + C + D + E + F + G? 174.________c_m_2 What is the area of the trapezoid shown, with top base of length 10cm and sides of lengths 10 cm and 6 cm? 175.________te_st_s The mean of Danielle’s test scores is 85. If Danielle’s lowest test score, which is 61, were to be discarded, the mean of her remaining test scores would be 88. How many tests did Danielle take? 176.______ve_r_tic_e_s This figure shows the net of a three-dimensional shape called a truncated octahedron. How many vertices does a truncated octahedron have? 177._______c_u_p_s Jamie is making pudding, using a recipe that calls for 1.5 cups of milk and 2 cups of flour. He has 7.75 cups of milk and would like to make a batch of pudding using all of the milk. How many cups of flour will he need in order to keep the ratio of ingredients constant? Express your answer as a decimal to the nearest tenth. 178._______y_e_ar_s When Shawna turned 21 years old, she was three times as old as Shelby. How old will old Shawna be when she is twice as old as Shelby? 179.__________ The sum of a number x and its reciprocal equals − 17 . What is the sum of all possible 4 values of x? Express your answer as a common fraction. 180.______ch_il_d_re_n There were 9 adults and 11 children at the movie at 11:45 a.m. By 11:50 a.m., 7 more adults and 8 more children were at the movie. At 12:00 p.m., there were 60 adults and children, and the ratio of adults to children was the same as at 11:45 a.m. How many more children came to the movie between 11:50 a.m. and 12:00 p.m.? 30 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Workout 5 181.__________ If each of the numbers 2, 7, 3 and 8 is assigned to one of the variables a, b, c and d, what is the greatest possible value of ab + bc + cd? 182._________cm_ If the volume of the air in an exercise ball is 137,250 cm3, what is the inside diameter of the ball? Express your answer to the nearest whole number. 183.__________ A square array of numbers is called an arithmetic square if each row 8 and column forms an arithmetic progression. If the remaining entries in 36 the array shown are filled in to create an arithmetic square, what is the greatest number in the array? 36 66 184._______u_n_its_2 6 A right triangle is inside a circle, with one vertex of the triangle at the 6 center of the circle and the other two vertices on the circle, as shown. If the circle has radius 6 units, what is the area of the shaded segment? Express your answer as a decimal to the nearest tenth. 185._______v_o_te_s Students at Baldwin Middle School cast 432 votes for student government president. If the number of votes the losing candidate received was 60% of the number that the winner received, how many more votes than the loser did the winner receive? 186._$_________ Michelle teaches on Tuesday and Thursday evenings. On Tuesdays she makes $112.50, and on Thursdays she makes $135. There are 31 days this month. What is the minimum amount that Michelle will earn this month? 187._______w_a_y_s Including the example shown, how many ways are there to completely cover this 2 × 7 grid with nonoverlapping 1 ×1 and 2 × 2 tiles, if rotations and reflections of these arrangements are considered distinct? 188.__________ If S1 = 4 and Sn = Sn–1 + (3 + n)n, what is the value of S3? 189.__________ The point P(7, 4) is reflected across the line ‫ ٴ‬to the point Ԣ(2, 6). What is the slope of line ‫ٴ‬ԝ? Express your answer as a common fraction. 190._______d_ro_p_s A veterinarian needs to prepare a vitamin mixture for parakeets. The ratio is 1 drop of vitamin oil to 75 drops of water. How many drops of vitamin oil are needed to prepare 1520 drops of the mixture? MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 31

Workout 6 191._______u_n_its_2 The triangle with vertices A(0, 4), B(3, 0) and C(4, 7) is dilated by a factor of 2 about the origin. What is the area of the dilated triangle? 192._______p_riz_e_s A raffle has prizes valued at $1, $3, $15, $60, $120, $360 and $1800. If exactly $10,000 in prizes is to be awarded, what is the least number of prizes that can be awarded, assuming that there will be at least one prize of each value awarded? 193.______p_ou_n_d_s The estimated weight of a salmon, in pounds, is the product of its length and the square of its girth, both in inches, divided by 775. What is the estimated weight of a salmon with length 28.5 inches and girth 10.25inches? Express your answer as a decimal to the nearest tenth. 194.__________ Point Q is the reflection of P(a, 4) over the x-axis, and R is the reflection of Q over the y-axis. If the area of %PQR is 12 units2 and a > 0, what is the value of a? Express your answer as a decimal to the nearest tenth. 195._______h_o_ur_s Trains pass through a rail yard on five separate tracks at five different frequencies:every 50 minutes, every hour, every 90 minutes, every 2 hours and every 3 hours. If trains pass through on all five tracks at 9:00 a.m., how many hours will elapse before trains next pass through on all five tracks at the same time? 196._______u_n_its_2 T S U R Quadrilateral PSTU is inscribed in semicircle O, as shown, with PQ= 3units, O QR=5 units and RS = 4 units. What is the area of quadrilateral PSTU? Express your answer as a decimal to the nearest tenth. Q P 197.________un_it_s Points A, B, C, D and E on a line are positioned so that B is the midpoint of segmentAD, D is three-fourths of the way from A to E, and C is one-fifth of the way from B to E. If AB= 6 units, what is the value of CE? 198._________%_ Vick pays $50 for 33 pounds of cashews, which he separates into smaller 3 -poundbags. 8 If Vick then sells each bag for $0.79, what percent of this price is profit? Express your answer to the nearest tenth of a percent. 199.__$________ To make bracelets, Alice bought 12 red beads and 10 white beads for $14, while Nevea bought 8red beads and 15 white beads for $13.50. What is the cost of a red bead? 200.__________ If 8 – x2 – y2 – 2xy = 0, for real numbers x and y, what is the value of |x + y|? Express your answer in simplest radical form. 32 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Competition Coach Toolkit This is a collection of lists, formulas and terms that Mathletes frequently use to solve problems like those found in this handbook. There are many others we could have included, but we hope you find this collection useful. Fraction Decimal Percent Common Arithmetic Series Prime Numbers ½ 0.5 50 1 + 2 + 3 + 4 + \" + n = n(n + 1) 2 43 ¹⁄3 0.3 33.3 2 3 47 ¼ 0.25 25 5 53 ¹⁄5 0.2 20 1 + 3 + 5 + 7 + \" + (2n − 1) = n2 7 59 ¹⁄6 0.16 16.6 11 61 ¹⁄8 0.125 12.5 2 + 4 + 6 + 8 + \" + 2n = n2 + n 13 67 ¹⁄9 0.1 11.1 17 71 ¹⁄10 0.1 10 Combinations & Permutations 19 73 ¹⁄11 0.09 9.09 23 79 ¹⁄12 0.083 8.3 n! n! 29 83 nCr = nPr = 31 89 37 97 r!(n − r)! (n − r)! 41 n n2 n3 Geometric Mean Divisibility Rules 111 ax and x = ab 2: units digit is 0, 2, 4, 6 or 8 248 x =b 3: sum of digits is divisible by 3 3 9 27 4: two-digit number formed by tens and 4 16 64 5 25 125 units digits is divisible by 4 6 36 216 5: units digit is 0 or 5 7 49 343 6: number is divisible by both 2 and 3 8 64 512 8: three-digit number formed by hundreds, 9 81 729 10 100 1000 tens and units digits is divisible by 8 11 121 1331 9: sum of digits is divisible by 9 12 144 1728 10: units digit is 0 13 169 2197 14 196 2744 15 225 3375 Equation of a Line Distance Traveled Quadratic Formula Distance = Rate × Time For ax2 + bx + c = 0, where a ă 0, Standard Form Ax + By = C x = −b ± b2 − 4ac 2a Pythagorean Triples Slope-Intercept Form (3, 4, 5) (5, 12, 13) (8, 15, 17) (9, 40, 41) y = mx + b (7, 24, 25) (12, 35, 37) m = slope b = y-intercept Point-Slope Form Difference of Squares Sum and Difference of Cubes a2 − b2 = (a + b) (a − b) y − y1 = m(x − x1) a3 − b3 = (a − b) (a2 + ab + b2) a3 + b3 = (a + b) (a2 − ab + b2) m = slope (x1, y1) = point on the line MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 33

Circles Area Given A(x1, y1) and B(x2, y2) π × r2 Distance from A to B = (x2 − x1)2 + (y2 − y1)2 Circumference 2×π×r=π×d For radius r Midpoint of AB = ⎛ x1 + x2 , y1 + y2 ⎛ ⎝ 2 2⎝ Arc Length Sector Area Slope of AB = y2 − y1 x2 − x1 x × 2 × π × r x × π × r2 360 360 For central angle Special Right Triangles of x degrees 30° 2a b2 45° Pythagorean Theorem a3 b ac a2 + b2 = c2 60° 45° b a b 30-60-90 45-45-90 Right Triangle Right Triangle Area of Polygons Square side length s s2 Polygon Angles (n sides) Rectangle length l, width w l×w Sum of the interior angle measures: 180 × (n − 2) Parallelogram base b, height h b×h Trapezoid bases b1, b2, 1 (b1 + b2) ×h Central angle measure of a regular polygon: height h 2 Rhombus diagonals d1, d2 1 × d1 × d2 360 2 n Triangle base b, height h 1 × b × h Interior angle measure of a regular polygon: 2 180 × (n − 2) 360 semi-perimeter s, n or 180 − n Triangle side lengths a, b, c s(s − a)(s − b)(s − c) Equilateral side length s s2 3 Triangle 4 Solid Dimensions Surface Area Volume Cube 6 × s2 s3 Rectangular side length s 2 × (l × w + w × h + l × h) Prism l×w×h Cylinder length l, width w, height h 2 × π × r × h + 2 × π × r2 π × r2 × h Cone circular base radius r, π × r2 + π × r × r2 + h2 height h 4 × π × r2 1 × π × r2 × h Sphere circular base radius r, 3 height h Pyramid 4 × π × r3 radius r 3 base area B, height h 1 × B × h 3 34 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Vocabulary & Terms The following list is representative of terminology used in the problems but should not be viewed as all-inclusive. It is recommended that coaches review this list with their Mathletes. absolute difference GCF (GCD) range of a function absolute value geometric sequence rate acute angle hemisphere ratio additive inverse (opposite) image(s) of a point(s) rational number adjacent angles ray apex (under a transformation) real number arithmetic mean improper fraction reciprocal (multiplicative inverse) arithmetic sequence infinite series reflection base ten inscribe regular polygon binary integer relatively prime bisect interior angle of a polygon revolution box-and-whisker plot intersection right angle center inverse variation right polyhedron chord irrational number rotation circ*mscribe isosceles scalene triangle coefficient lateral edge scientific notation collinear lateral surface area sector common divisor lattice point(s) segment of a circle common factor LCM segment of a line common fraction median of a set of data semicircle complementary angles median of a triangle semiperimeter congruent mixed number sequence convex mode(s) of a set of data set coordinate plane/system multiplicative inverse significant digits coplanar similar figures counting numbers (reciprocal) slope counting principle natural number space diagonal diagonal of a polygon obtuse angle square root diagonal of a polyhedron ordered pair stem-and-leaf plot digit sum origin supplementary angles dilation palindrome system of equations/inequalities direct variation parallel tangent figures divisor Pascal’s Triangle tangent line domain of a function percent increase/decrease term edge perpendicular transformation equiangular planar translation equidistant polyhedron triangular numbers expected value polynomial trisect exponent prime factorization twin primes exterior angle of a polygon principal square root union factor proper divisor unit fraction finite proper factor variable frequency distribution proper fraction whole number frustum quadrant y-intercept function quadrilateral random range of a data set MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 35

PROBLEM INDEX It is very difficult to categorize many of the problems in this handbook. MATHCOUNTS problems often straddle multiple categories and cover several concepts, but in this index, we have placed each problem in exactly one category and mapped it to exactly one Common Core State Standard (CCSS). In this index, code 9 (3) 7.SP.3 would refer to problem #9 with difficulty rating 3 mapped to CCSS 7.SP.3. The difficulty rating and CCSS mapping are explained below. DIFFICULTY RATING: Our scale is 1-7, with 7 being most difficult. These general ratings are only approximations: • 1, 2 or 3: Appropriate for students just starting the middle school curriculum; 1 concept; 1- or 2-step solution. • 4 or 5: Knowledge of some middle school topics necessary; 1-2 concepts; multi-step solution. • 6 or 7: Knowledge of advanced middle school topics and/or problem-solving strategies necessary; multiple and/or advanced concepts; multi-step solution. COMMON CORE: We align our problems to the NCTM Standards for Grades 6-8, however we also have mapped these problems to CCSS because 42 states, D.C., 4 territories and the Dept. of Defense Education Activity (DoDEA) have voluntarily adopted it. Our CCSS codes contain (in this order): 1. Grade level in the K-8 Standards for Mathematical Sd Pg Content (SMC). Courses that are in the high school SMC instead have the first letter of the course name. STATISTICS PLANE GEOMETRY 2. Domain within the grade level or course and then the 11 (3) 6.SP.5 individual standard. 12 (2) 6.SP.4 34 (2) 8.G.5 16 (3) 6.SP.2 59 (3) 8.G.5 Here are 2 examples: 17 (3) 6.SP.5 75 (3) SMP • 6.RP.3 o Standard #3 in the Ratios and Proportional 18 (4) 7.SP.3 82 (3) SMP Relationships domain of grade 6 74 (3) 7.SP.3 92 (3) 7.G.6 • G-SRT.6 o Standard #6 in the Similarity, Right Triangles 83 (2) 6.SP.2 110 (5) G-SRT.6 and Trigonometry domain of Geometry 96 (3) 6.SP.2 114 (3) 8.G.7 143 (2) 6.SP.2 140 (5) 7.G.6 Some math concepts are not specifically mentioned in CCSS. 149 (2) 6.SP.4 150 (3) 7.G.5 For problems using these concepts, we use the code of a related 158 (2) 6.SP.5 159 (3) 7.G.6 standard, when possible. Some of our problems are based on 161 (2) 7.NS.2 165 (4) SMP concepts outside the scope of CCSS or are based on concepts 170 (3) 6.SP.5 184 (5) G-C.2 in the K-5 SMC but are more difficult than a grade K-5 problem. 175 (3) 7.SP.3 196 (6) G-SRT.6 When appropriate, we coded these problems SMP for the CCSS Standards for Mathematical Practice. Cg Sg Me Pr COORDINATE SOLID GEOMETRY GEOMETRY MEASUREMENT PROPORTIONAL REASONING 53 (1) 5.G.4 32 (1) 7.G.3 44 (2) 5.MD.1 63 (1) 8.G.8 102 (2) 7.G.6 67 (3) 7.G.6 40 (2) 6.RP.2 118 (3) 8.F.4 120 (4) 8.G.9 79 (4) 7.G.6 57 (3) 7.RP.1 156 (4) 7.G.6 126 (5) 7.G.6 116 (4) SMP 88 (2) 6.RP.3 189 (4) G-GPE.5 138 (4) 8.G.9 121 (2) 7.G.6 146 (2) 6.RP.1 191 (5) 8.G.3 176 (5) 7.G.6 163 (4) G-SRT.6 147 (1) 7.RP.1 194 (4) 7.G.6 174 (3) 7.G.6 151 (3) 6.RP.3 182 (3) G-GMD.3 167 (4) 6.RP.3 193 (2) 6.EE.7 177 (3) 6.RP.3 180 (4) SMP 190 (2) 6.RP.3 50 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

Pf Nt Al Pc PERCENTS & NUMBER THEORY ALGEBRAIC PROBABILITY, FRACTIONS EXPRESSIONS & COUNTING & 33 (1) 6.NS.2 COMBINATORICS 1 (1) 6.RP.3 43 (2) 6.NS.4 EQUATIONS 2 (2) 6.RP.3 54 (2) 6.EE.1 24 (3) S-CP.9 3 (1) 6.RP.3 56 (2) SMP 26 (4) A-APR.5 29 (4) 7.SP.8 4 (3) 6.RP.3 58 (3) 6.NS.4 27 (4) A-APR.5 36 (1) 7.SP.7 5 (4) 6.RP.3 60 (4) SMP 28 (5) A-APR.5 50 (3) S-CP.9 6 (4) 6.RP.3 77 (1) 7.NS.2 38 (2) 6.EE.2 76 (2) 7.SP.6 7 (5) 6.RP.3 80 (4) SMP 39 (2) 6.EE.6 97 (4) S-CP.6 8 (4) 6.RP.3 90 (3) 7.NS.3 47 (2) 6.EE.6 100 (4) S-CP.9 9 (3) 6.RP.3 93 (2) 8.EE.4 70 (3) 6.EE.9 111 (4) S-CP.6 10 (4) 7.RP.1 98 (4) A-SEE.2 73 (2) 7.NS.2 113 (4) 7.SP.8 13 (3) 6.RP.1 104 (3) 8.NS.1 81 (2) 6.EE.1 123 (3) 7.SP.8 15 (4) 7.NS.3 106 (2) 6.EE.1 84 (3) 7.EE.4 128 (4) S-CP.7 19 (3) 6.RP.3 109 (4) SMP 87 (3) 6.EE.2 130 (4) SMP 20 (5) 7.RP.3 115 (4) N-RN.2 95 (3) 8.EE.8 134 (3) S-CP.9 23 (2) SMP 125 (3) N-RN.2 103 (1) 6.EE.2 172 (3) S-CP.9 45 (1) 6.NS.1 132 (3) SMP 105 (3) 8.F.2 187 (5) S-CP.9 49 (2) 7.RP.3 133 (5) N-RN.2 112 (2) 8.EE.7 66 (2) 7.RP.3 139 (4) SMP 119 (2) SMP Ps 69 (3) 6.NS.1 141 (2) 6.NS.4 122 (3) 8.F.1 78 (3) 7.RP.3 148 (2) 7.NS.1 129 (4) A-REI.4 PROBLEM SOLVING 91 (1) 6.RP.3 152 (3) 6.EE.9 131 (3) 8.F.1 (MISCELLANEOUS) 94 (1) 6.NS.1 155 (3) 6.NS.4 136 (4) 8.EE.2 124 (3) SMP 160 (3) 6.NS.4 142 (1) SMP 21 (3) SMP 145 (2) 6.RP.3 166 (3) 6.NS.4 164 (2) 7.NS.3 22 (2) SMP 154 (3) 6.RP.3 168 (1) 6.NS.4 171 (3) 6.EE.6 25 (3) SMP 185 (4) 6.RP.3 192 (4) SMP 173 (4) 8.EE.8 30 (7) S-CP.9 197 (3) 7.NS.2 195 (4) 6.RP.3 178 (3) 7.EE.4 41 (1) SMP 198 (4) 7.NS.3 179 (4) A-REI.4 48 (3) 6.NS.1 Lo 199 (4) 7.EE.4 51 (1) SMP Gm 200 (5) A-REI.2 86 (3) SMP LOGIC 89 (3) 7.EE.4 GENERAL MATH Sp 107 (3) SMP 14 (5) SMP 117 (4) 6.EE.7 31 (1) 6.NS.3 64 (2) F-BF.2 SEQUENCES, SERIES 127 (3) 6.NS.4 35 (1) 3.OA.2 68 (3) SMP & PATTERNS 135 (3) SMP 37 (1) 5.NBT.2 99 (4) SMP 137 (3) SMP 42 (2) 7.NS.1 162 (1) SMP 46 (2) F-IF.3 144 (2) 7.RP.1 52 (1) 7.NS.1 181 (4) SMP 62 (1) F-IF.3 153 (2) SMP 55 (1) 6.NS.3 186 (4) SMP 108 (3) F-BF.2 157 (2) 7.NS.3 61 (1) 7.RP.1 183 (3) F-BF.2 169 (4) 7.NS.3 65 (2) 3.OA.8 188 (3) F-IF.3 71 (2) SMP 72 (1) 6.NS.3 COACHES: 85 (2) 8.EE.1 FIND PROBLEMS, ANSWERS, SOLUTIONS 101 (2) 6.NS.1 + PROBLEM INDEX FOR PROBLEMS 201–250 at www.mathcounts.org/coaches! MATHCOUNTS 2020-2021 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. 51

ANSWERS MIXTURE STRETCH WARM-UP 2 WARM-UP 6 WARM-UP 10 WORKOUT 3 41. 4:12 (1) 81. 6 (2) 121. 24 (2) 161. 1.75 1. 1* (1) 42. 19 (2) 82. 1/2 (3) 122. −50 (3) 162. 35 (2) 43. 60 (2) 83. 12 (2) 123. 7/20 (3) 163. 13.9 (1) 2. 7/20 (2) 44. 11 (2) 84. 111 (3) 124. 3/5 (3) 164. 160 (4) 45. 10 (1) 85. 0 (2) 125. 1/3 (3) 165. 12 (2) 3. 3/7 (1) 46. 57 (2) 86. 3/16 (3) 126. 72 (5) 166. 10,101 (4) 47. −9 (2) 87. 4 (3) 127. 10,467 (3) 167. 326,000 (3) 4. 14 (3) 48. 72 (3) 88. 10 (2) 128. 5/12 (4) 168. 33 (4) 49. 6 (2) 89. 24 (3) 129. 8 (4) 169. 70 (1) 5. 8 (4) 50. 17 (3) 90. 10 (3) 130. 34 (4) 170. 1.2 (4) (3) 6. 1.74 (4) 7. 1.25 (5) 8. 4 (4) 9. 2 (3) 10. 5.2 (4) STATISTICS WARM-UP 3 WARM-UP 7 WARM-UP 11 WORKOUT 4 (3) STRETCH (3) 51. 5 (1) 91. 28 (1) 131. 10 (3) 171. −4/7 (3) 11. 73 (2) 52. 10 (1) 92. 100 (3) 132. 199 (3) 172. 5760 (4) 12. 3216 (3) 53. 7 (1) 93. 200 (2) 133. 27 (5) 173. 1 (3) 13. 5/8 (5) 54. 1/9 (2) 94. 22 (1) 134. 720 (3) 174. 84 (3) 14. 36 (4) 55. 6.666 (1) 95. 60 (3) 135. 4 (3) 175. 9 (5) 15. 13/22 (3) 56. 15 (2) 96. 12 (3) 136. 2 (4) 176. 24 (3) 16. 29 (3) 57. 3 (3) 97. 1/3 (4) 137. 6 (3) 177. 10.3 (3) 17. 2.36 (4) 58. 15 (3) 98. 40 (4) 138. 2ξ21 (4) 178. 28 (4) 18. 42 (3) 59. 58 (3) 99. 12 (4) 139. 63 (4) 179. −17/4 (4) 19. 60 (5) 60. 37 (4) 100. 28,800 (4) 140. 72/5 (5) 180. 14 20. 18.3 PASCAL’S TRIANGLE STRETCH WARM-UP 4 WARM-UP 8 WORKOUT 1 WORKOUT 5 21. 6435 (1) 101. 12 (2) 141. 270 (3) 61. 42 (1) 102. 36 (2) 142. 10 (2) 181. 94 (4) (1) 103. 18 (1) 143. 356 22. 7 (2) 62. 79 (2) 104. 11/7 (3) 144. 0.31 (1) 182. 64 (3) (2) 105. 1/3 (3) 145. 59 23. 4096 (2) 63. 5 (2) 106. 2 (2) 146. 720 (2) 183. 79 (3) (3) 107. 19 (3) 147. 1115 24. 56 (3) 64. 25 (3) 108. 27 (3) 148. 289 (2) 184. 10.3 (5) (3) 109. 4 (4) 149. 4.5 25. 220 (3) 65. 98 (3) 110. 2ξ6 (5) 150. 54 (2) 185. 108 (4) 26. 1792 (4) 66. 40 (2) 186. 990 (4) 27. 81 (4) 67. 9 7 (1) or 990.00 20 28. 1011 (5) 68. 4 (2) 187. 21 (5) 29. 11/16 (4) 69. 4/3 (2) 188. 245 (3) 30. 8 (7) 70. 4 (3) 189. 5/2 (4) 190. 20 (2) WARM-UP 1 WARM-UP 5 WARM-UP 9 WORKOUT 2 WORKOUT 6 (5) 31. 872,000 (1) 71. 111 (2) 111. 19/66 (4) 151. 1.64 (3) 191. 25 (4) 32. 27 (1) 72. 59/50 (1) 112. 2 (2) 152. 1* (3) 192. 16 (2) 33. 0 (1) 73. 10/11 (2) 113. 1/16 (4) 153. 9.41 (2) 193. 3.9 (4) 34. 100 (2) 74. 80 (3) 114. ξ10 (3) 154. 125 (3) 194. 1.5 (4) 35. 13 (1) 75. 5/18 (3) 115. 8 × 1021 (4) 155. 216,000 (3) 195. 30 (6) 36. 5/9 (1) 76. 1/4 (2) 116. 1 (4) 156. 12 (4) 196. 46.2 (3) 37. 10 (1) 77. −4 (1) 117. 25 (4) 157. 14.25 (2) 197. 8 (4) 38. 67 (2) 78. 3 or 3.00 (3) 118. 9/2 (3) 158. 504 (2) 198. 28.1 (4) 39. 6 (2) 79. 4 (4) 119. 311 (2) 159. 7.5 (3) 199. 0.75 (5) 40. 8 (2) 80. 0 (4) 120. 81π (4) 160. 7 (3) 200. 2ξ2 * The plural form of the units is always provided in the answer blank, even if the answer appears to require the singular form of the units. 52 Copyright MATHCOUNTS, Inc. 2020. All rights reserved. MATHCOUNTS 2020-2021

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