2022 School competition B - mathcounts.org

Copyright MATHCOUNTS, Inc. 2021. All rights reserved.

Founding SponSorS: National Society of Professional Engineers, National Council of Teachers of Mathematics and CNA Insurance

www.mathcounts.org/competitioncoaches

2022 School competition Booklet

TiTle SponSorS

Raytheon TechnologiesU.S. Department of Defense STEM

lead SponSor

Northrop Grumman Foundation

naTional SponSorS

National Society of Professional Engineers3Mgives

Texas Instruments IncorporatedArt of Problem Solving

competition coach ReSouRceS

School Competition Booklet

Countdown Round PowerPoint®

Competition Updates & Edits

www.mathcounts.org/competitioncoaches

GENERAL INSTRUCTIONS

This section contains instructions, rules and procedures for administering the MATHCOUNTS School Competition. It is important that the coach look upon coaching sessions during the academic year as opportunities to develop better mathematics skills in all participants. Therefore, the coach is encouraged to postpone the selection of those students who will be competing at the Chapter Competition until just prior to the event in February. Selection of the students need not be based on the results of a school competition. For this reason, schools may deviate from these rules in administering the School Competition. However, experience with the official rules may aid the students who compete at the chapter, state and national levels.

Individual scores are kept for each participating student, and a team score is calculated for each team. Before beginning the competition, divide participants into teams of four students and designate a captain for each team.

At the end of each round of the competition, collect all competition booklets, problems and scratch paper.

1. Use of notes and other aids (including graph paper, rulers, compasses, protractors, reference tables, and dictionaries) is not permitted.

2. The use of connected wearable technology and “smart” devices including, but not limited to cellular phones, smartphones, laptops, tablets, e-readers, calculator wrist watches and smartwatches is prohibited. Students are not permitted to access such devices during any round of the competition.

3. The Target and Team Rounds assume the use of a calculator. Calculator use is permitted in these rounds only. Any calculator that does not contain a QWERTY (i.e., typewriter-like) keypad is permitted. Calculators that have the ability to enter letters of the alphabet but do not have a keypad in a standard typewriter arrangement are acceptable. Students may not use calculators to exchange information with another person or device during the competition.

4. Talking and signals are permitted only during the Team Round.5. Before the competition, coaches should review with students the rules for acceptable forms of

answers, found in the SCORING section of this booklet.

SPRINT ROUND INSTRUCTIONS

1. Distribute scratch paper. 2. Distribute Sprint Round booklet.3. Instruct each student to print his/her name on the front of the booklet, in the allotted space.4. Instruct each student to write the digits 0 through 9 on the booklet, in the spaces provided. This

may be used during scoring to clarify written answers that are not legible.5. Read aloud instructions printed on the front of the booklet while students read instructions silently.6. Instruct students to begin. Start timing.7. After 37 minutes, give a three-minute warning. 8. After 40 minutes, say, “Stop, pencils down,” and instruct students to close competition booklets.

TARGET ROUND INSTRUCTIONS

1. Distribute scratch paper. 2. Distribute the first (next) pair of Target Round problems, and instruct each student to print his/her

name in the allotted space.3. Read aloud the instructions printed on the cover of the first pair of problems.4. Instruct students to begin. Start timing.5. Give a 30-second warning at 5 minutes 30 seconds. 6. After 6 minutes, say, “Stop, pencils down.”7. Collect all papers.8. For each of the next three pairs of problems, repeat step 2, then steps 4 through 7.

TEAM ROUND INSTRUCTIONS

1. Arrange all teams (of four students) in a room with at least five feet of unoccupied space between teams.

2. Distribute scratch paper.3. Distribute Team Round booklet to each person, and instruct team captain to print team name and

team members’ names on his/her booklet. This becomes the team’s official answer booklet.4. Read aloud the instructions printed on the cover of the booklet while students read the instructions

silently.5. Instruct students to begin. Start timing.6. After 17 minutes, give a three-minute warning. 7. After 20 minutes, say, “Stop, pencils down,” and instruct students to close competition booklets.

COUNTDOWN ROUND INSTRUCTIONS

The Countdown Round is a mandatory component at the National Competition, and it is used to determine the final rank of the top competitors. At the chapter and state levels, the use of the Countdown Round, officially or unofficially, is at the discretion of the state coordinator. When used officially, the Countdown Round will adhere to the rules presented below. The instructions may be modified as necessary at the school level. This round is available only in PowerPoint® format for the School Competition. The PowerPoint file can be downloaded by logging in at www.mathcounts.org/competitioncoaches and then accessing the Coach Resources.

1. Based on scores in the Sprint and Target Rounds, rank all competitors and select the top 25%, up to ten students, to compete in the Countdown Round.

2. Seat the two lowest-ranked students so they are in clear view of the moderator. Each competitor should be given scratch paper and sharpened pencils before the round begins. Invite the competitors to introduce themselves and, if applicable, test their buzzers.

3. Read the following statement to all students who will be competing in the round:

In this round, I will read each problem aloud as it is presented to you. You may use the scratch paper and pencil in front of you to calculate your answer to the problem. You are not allowed to use calculators during this round.

You will have a maximum of 45 seconds to solve the problem after it is presented. You will be given a ten-second warning before time expires. As soon as you have solved the problem, press your buzzer. [Schools may have alternate methods of determining order of finish and should adjust directions to students accordingly.] I will call on the first person who signals. Do not announce your answer until I call on you. Each time you wish to answer, you must signal, though you may not answer more than once for any question. If you do not signal before you answer, your answer will be disqualified. If you answer after signaling but before I call on you, your answer will be accepted, but I ask that you please wait until you hear your name so that there is no confusion.

Once I call on you, you will have three seconds to begin your answer. Your opponent may continue working while you are responding.

If you answer correctly, you will score one point in the round. If you answer incorrectly, your opponent will have the remainder of the allotted 45 seconds to press his/her buzzer for an opportunity to answer the problem and score a point in the round.

Whoever answers the most of the three problems correctly (not necessarily two out of the three) will progress to the next round to compete for the next place. If you are tied after three questions, I will declare a sudden victory situation. I will describe the rules for this process should this situation arise.

It is very important that rules be followed exactly. If you answer without signaling your buzzer, your answer will be disqualified. Are there any questions?

[Note that the above procedure does not necessarily require a student to answer two out of the three problems correctly. For instance, a student answering only one problem of three will progress to the next round if his/her opponent has not correctly answered any questions in the round.]

4. Conduct the round as described above. After the winner of each round is identified, dismiss his/her opponent, and ask the next written competition place-holder to be seated to participate in the next round. Invite the new competitor to introduce himself or herself and, if applicable, test his/her buzzer.

5. If a sudden victory situation occurs, read the following statement to the students:

Since you are tied at the end of three problems, I must declare a sudden victory situation. I will now continue to read problems to both of you. Rules for answering problems remain the same as before. The first one of you to answer a problem correctly will advance to the next round.

6. After the winner of this round is identified, congratulate the winner, have the two competitors shake hands and call for applause for the student who has lost and is leaving the stage. Continue in this manner after the winner of each round is identified.

7. Just before the 4th-ranked student competes in his/her first round, read the following statement to the students:

For the final four rounds, our rules will change slightly. In order to win a round, our Mathletes will have to answer three problems correctly. The first Mathlete in each round to answer three problems correctly will progress to the next round.

8. Repeat procedure until the Champion of the Countdown Round is identified.

*Rules for the Countdown Round change for the National Competition.

SCORING

1. The following rules explain acceptable forms for answers.a. Units of measurement are not required in answers, but they must be correct if given. When a problem

asks for an answer expressed in a specific unit of measure or when a unit of measure is provided in the answer blank, equivalent answers expressed in other units are not acceptable. For example, if a problem asks for the number of ounces and 36 oz is the correct answer, 2 lbs 4 oz will not be accepted. If a problem asks for the number of cents and 25 cents is the correct answer, $0.25 will not be accepted.

b. All answers must be expressed in simplest form. A “common fraction” is to be considered a fraction in the form ± a

b , where a and b are natural numbers and GCF(a, b) = 1. In some cases the term “common fraction” is to be considered a fraction in the form A

B , where A and B are algebraic expressions and A and B do not share a common factor. A simplified “mixed number” (“mixed numeral,” “mixed fraction”) is to be considered a fraction in the form ± N a

b , where N, a and b are natural numbers, a < b and GCF(a, b) = 1.

c. Ratios should be expressed as simplified common fractions unless otherwise specified.

d. Radicals must be simplified. A simplified radical must satisfy: 1) no radicands have a factor which possesses the root indicated by the index; 2) no radicands contain fractions; and 3) no radicals appear in the denominator of a fraction. Numbers with fractional exponents are not in radical form.

e. Answers to problems asking for a response in the form of a dollar amount or an unspecified monetary unit (e.g., “How many dollars...,” “How much will it cost...,” “What is the amount of interest...”) should be expressed in the form ($) a.bc or a.bc (dollars), where a is an integer and b and c are digits. The only exceptions to this rule are when a is zero, in which case it may be omitted, or when b and c are both zero, in which case they both may be omitted. Answers in the form ($)a.bc or a.bc (dollars) should be rounded to the nearest cent unless otherwise specified.

f. Do not make approximations for numbers (e.g., π, 23 , 5 3 ) in the data given or in solutions unless the

problem says to do so.

g. Do not perform any intermediate rounding (other than the “rounding” a calculator performs) when calculating solutions. All rounding should be done at the end of the calculation process.

h. Scientific notation should be expressed in the form a × 10n where a is a decimal, 1 < |a| < 10, and n is an integer.

i. An answer expressed to a greater or lesser degree of accuracy than called for in the problem will not be accepted. Whole number answers should be expressed in their whole number form. Thus, 25.0 will not be accepted for 25, and 25 will not be accepted for 25.0.

j. The plural form of the units will always be provided in the answer blank, even if the answer appears to require the singular form of the units.

2. Specific instructions stated in a given problem should take precedence over any general rule or procedure.

3. Scores are kept for individuals and teams. The individual score is the number of Sprint Round questions answered correctly plus two times the number of Target Round questions answered correctly. The maximum possible individual score is 46. The team score is calculated by dividing the sum of the team members’ individual scores by four, and then adding two times the number of Team Round questions answered correctly. The maximum possible team score is 66.

Because the School Competition is not the only mechanism available to determine which students should advance to the Chapter Competition, ties on the School Competition needn’t be broken. At the Chapter, State and National Competitions, however, ties among individuals or teams will be broken by comparing the scores of specific rounds.

Copyright MATHCOUNTS, Inc. 2021. All rights reserved. 2022 School Answer Key

Sprint Round Answers

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27. 28.

29.

30.

9

64

30 degrees

8

25 percent

141

10,000 seats

55

7

$750 or 750.00

17

8

3

8

27 dots

$18 or 18.00

507

16

3

17 inches

6

9

63 integer values

867

10

434

30,000

31 cm2

5

3

8

Target Round Answers

Team Round Answers

3.

4.

1.

2.

5.

6.

7.

8.

2:35 p.m.

9

6

98 inches 4 yd/s

45 degrees

9 marbles

1.

2.

3.

4.

5.

6. 7.

8.

9. 10.

30%

40 ounces

8 ordered pairs

319

24 comic books

32 cm

18

3 2 inches

7 zeros

3

1

15

7

16

Total Correct Scorer’s Initials


Name

DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.

This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You are not allowed to use calculators, books or other aids during this round. Calculations may be done on scratch paper. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left-hand column of the competition booklet. If you complete the problems before time is called, use the remaining time to check your answers.

In each written round of the competition, the required unit for the answer is included in the answer blank. The plural form of the unit is always used, even if the answer appears to require the singular form of the unit. The unit provided in the answer blank is the only form of the answer that will be accepted.


2022 School competition

Sprint Round Problems 1−30

TiTle SponSorS


lead SponSor


naTional SponSorS



1

2

3

4

5

6

7

8

9

0

Copyright MATHCOUNTS, Inc. 2021. All rights reserved. 2022 School Sprint Round

1. _____________

2. _____________

3. _____________

4. _____________

5. _____________

What is the median of the composite integers that are greater than 1 and less than 15?

The square root of what number is double the value of 4?

Square BCDE and equilateral triangle ABC are coplanar and share side CB, as shown. What is the degree measure of angle ACD?

Given that (x, y) = (3, 6), what is the value of y2 − 2xy + 8?

A pair of boots that originally cost $96 is on sale for $72. By what percent has the original cost been reduced?

C

B E

D

A

degrees

percent


6. _____________

7. _____________

8. _____________

9. _____________

10. _____________

What is the sum of the mean, median and mode of the numbers in the set {46, 52, 46, 58, 43}?

A sports arena has a seating capacity of 21,000 seats. One-third

of the seats are box seats, and 2

7 of the remaining seats are

reserved seats. All other seats are for general admission. How

many seats are for general admission?

What is the value of one-half the sum of the first 10 positive even integers?

If a rectangle has vertices (3, −2), (−1, 4) and (−1, −2), what is the sum of the coordinates of the fourth vertex?

A farmer plants seeds for a 60-acre field of green beans. A 20-pound bag of seed costs $25, and 10 pounds of seed are needed per acre. How much will it cost the farmer to seed this field?

ReservedBox

General

seats

$


11. _____________

12. _____________

13. _____________

14. _____________

15. _____________

A bag contains 13 yellow, 10 blue, 5 red and 4 green tokens. What is the probability that a randomly chosen token is blue or green? Express your answer as a common fraction.

Twenty-two chairs, numbered consecutively from 1 to 22, are evenly spaced around a circular table. What is the number of the chair that is located directly across from chair number 6?

Three numbers are in the ratio 2:4:10 and have a sum of 64. What is the least of the three numbers?

Two circles with radii of 3 cm and 4 cm are externally tangent. A third circle is circumscribed about the first two circles as shown. What is the ratio of the area of the smallest circle to the area of the shaded region? Express your answer as a common fraction.

In the pattern shown, each triangle after the first has one more dot per side than the previous triangle. If the pattern continues, what is the total number of dots in the triangle with 10 dots per side?

dots


16. _____________

17. _____________

18. _____________

19. _____________

20. _____________

After Josie spent one-half of her money on a book and gave away one-third of the remainder, she was left with exactly enough money for subway fare and a soda. If the subway fare was three times the price of the soda, which was $1.50, how much money did Josie have initially?

What is the sum of the numbers of vertices, edges and faces of the octagonal prism shown?

Each of the triangles in the figure shown is equilateral. What fraction of the entire figure is shaded? Express your answer as a common fraction.

The line given by the equation y = mx + 8 contains the point (−1, 5). What is the value of m?

Simon has three ropes whose lengths are 51 inches, 68 inches and 85 inches. He wants to cut the ropes into pieces of equal length with no rope wasted. What is the greatest possible length of each piece?

$

inches


21. _____________

22. _____________

23. _____________

24. _____________

25. _____________

Given that the sum of the values of the letters of MATHCOUNTS is 11 and the value of each consonant is three more than the value of each vowel, what is the sum of the values of the letters of COUNTS?

February 22, 2022, written in the format MM/DD/YYYY, is 02/22/2022, which contains only two distinct digits. Consider the next date after 02/22/2022

that contains only two distinct digits when written in this format. What is the sum of all eight digits of that date?

For how many integer values of x is it true that 13 < x2

< 45?

Four consecutive even integers sum to 332. What is the greatest of the four integers?

A box contains five balls numbered 1, 2, 3, 4 and 5. Three balls are randomly selected without replacement. What is the probability that the median of the values on the selected balls is less than 4? Express your answer as a common fraction.

integervalues


26. _____________

27. _____________

28. _____________

29. _____________

30. _____________

The sum of three distinct prime numbers is 40. What is their product?

What is the sum of the 100 smallest positive integers that are multiples of 3 but that are not multiples of 6?

The lengths indicated for the rectangle shown are in centimeters. What is the area of the shaded region?

Let x ⧪ y = 2xy . Suppose (a ⧪ b)⧪c − a ⧪ (b ⧪ c) = a, where a, b and c are

nonzero integers. What is the sum of all possible positive values of c?

An unfair coin has probability p of landing heads up on a single flip, where 0 < p < 1. The coin is tossed seven times, and the probability that the coin lands heads up exactly twice is equal to the probability that the coin lands heads up exactly three times. What is the value of p? Express your answer as a common fraction.

66

5

5

9

9

2

2

cm2


Name


This section of the competition consists of eight problems, which will be presented in pairs. Work on one pair of problems will be completed and answers will be collected before the next pair is distributed. The time limit for each pair of problems is six minutes. The first pair of problems is on the other side of this sheet. When told to do so, turn the page over and begin working. This round assumes the use of calculators, and calculations also may be done on scratch paper, but no other aids are allowed. All answers must be complete, legible and simplified to lowest terms. Record only final answers in the blanks in the left-hand column of the problem sheets. If you complete the problems before time is called, use the time remaining to check your answers.

Scorer’s InitialsProblem 1 Problem 2



Target Round Problems 1 & 2

TiTle SponSorS


lead SponSor


naTional SponSorS



1

2

3

4

5

6

7

8

9

0

Copyright MATHCOUNTS, Inc. 2021. All rights reserved. 2022 School Target Round

1. _______________

2. _______________

Alex drives from Anytown to Some Place. Alex arrives in Some Place at 9:12 p.m. If Alex’s trip took 6 hours 37 minutes, at what time did Alex depart Anytown?

: p.m.

SomePlace

What is the 100th digit after the decimal point in the decimal representation

of 1

13?




Name




TiTle SponSorS


lead SponSor


naTional SponSorS



1

2

3

4

5

6

7

8

9

0


3. _____________

4. _____________

The three vertices of a triangle lie on a coordinate plane at the points (2, 0), (4, 0) and (a, b), where b > 0. The area of the triangle is 6 units2. What is the value of b?

A cylinder with a radius of 2 inches is full of water. The water is poured into a different cylinder with a radius of 7 inches and a height of 8 inches. After all the water is poured in the second cylinder, it is full. What is the height of the first cylinder?

inches


Name






TiTle SponSorS


lead SponSor


naTional SponSorS



1

2

3

4

5

6

7

8

9

0


5. _____________

6. _____________

Two different prime numbers are selected at random from among the first ten prime numbers. What is the probability that the sum of the two primes is 30? Express your answer as a common fraction.

Kesia runs at a constant rate so that she covers d yards in t seconds.

If the distance is increased by x yards, the time required is increased

by 1

4x seconds. What is Kesia’s rate in yards per second?

yd/s




Name




TiTle SponSorS


lead SponSor


naTional SponSorS



1

2

3

4

5

6

7

8

9

0


7. _____________

8. _____________

In triangle ABC, AX = XY = YB = BC and mABC = 120 degrees. What is the

degree measure of angle ACB?

degrees

A

B

CX

Y

Fifteen red and fifteen blue marbles are placed in a jar. Twenty-four marbles are randomly removed from the jar and put in a second jar. What is the absolute difference between the number of red marbles in the second jar and the number of blue marbles remaining in the first jar?

marbles


TeamMembers , Captain


This section of the competition consists of 10 problems which the team has 20 minutes to complete. Team members may work together in any way to solve the problems. Team members may talk to each other during this section of the competition. This round assumes the use of calculators, and calculations also may be done on scratch paper, but no other aids are allowed. All answers must be complete, legible and simplified to lowest terms. The team captain must record the team’s official answers on his/her own competition booklet, which is the only booklet that will be scored. If the team completes the problems before time is called, use the remaining time to check your answers.



Team Round Problems 1−10

Total Correct Scorer’s Initials

TiTle SponSorS


lead SponSor


naTional SponSorS



1

2

3

4

5

6

7

8

9

0

Copyright MATHCOUNTS, Inc. 2021. All rights reserved. 2022 School Team Round

1. _____________

2. _____________

3. _____________

4. _____________

5. _____________

A group of students was asked to vote on whether their favorite farm animal is a chicken, a cow, a horse, a pig or some other animal. All of the voting results are shown in this histogram. What percent of the students voted for a cow?

A container is one-half full. After 20 ounces are poured out, the container is one-third full. How many ounces are still in the container?

How many ordered pairs of positive integers whose greatest common factor is 1 have a sum of 30? For example, the ordered pair (5, 25) cannot be included, but the ordered pair (19, 11) is one to included.

Max and Lucy are playing twenty questions with numbers. Max thought of a number and Lucy is asking yes-or-no questions to figure out what number it is. Based on the questions and answers shown, what is Max’s number?

James and Joe started with the same number of comic books. James gave 12 of his comic books to Joe. Now, Joe has three times as many comic books as James has. How many comic books did James have at the start?

%

Favorite Farm Animal Voting Results

?% o

f Stu

dent

s

Farm AnimalsChicken Cow Horse Pig Other

15%

25%30

10

20

10%

20%

ounces

orderedpairs

Lucy’s Question Max’s AnswerIs the number an integer? YesIs the number less than 100? NoIs the number less than 500? YesDoes the number have any digits that are even? NoIs the number divisible by 3? NoIs the number divisible by 11? Yes

comicbooks

Copyright MATHCOUNTS, Inc. 2021. All rights reserved. 2022 School Team Round

6. _____________

7. _____________

8. _____________

9. _____________

10. _____________

In the figure shown, each vertex is the intersection of two perpendicular segments. What is the perimeter of this figure?

What is the mean of a set of five numbers if the three smallest numbers have a mean of 12 and the two largest numbers have a mean of 27?

A square and an equilateral triangle have equal perimeters. The area of the triangle is 4 3 in2. What is the length of the diagonal of the square? Express your answer in simplest radical form.

How many zeros are to the right of the last nonzero digit of 30!?

What is the remainder when 20212021 is divided by 7?

6 cm

7 cm

3 cm

cm

inches

zeros

2022 School competition B - mathcounts.org

Documents

Transcript of 2022 School competition B - mathcounts.org