Ze Committee ZeMC

Ze Committee ZeMC

ZeMC 8
Ze Committee Ze Math Competition

2nd Annual
DO NOT OPEN UNTIL Sunday, January 1st, 2023

**Administration will disqualify your school’s results.** ZeMC 8

• None of the information needed to administer this competition is contained in Sunday, January 1st, 2023
the ZeMC 8 Teacher’s Manual. PLEASE DO NOT READ THE MANUAL AS
IT DOES NOT EXIST. INSTRUCTIONS

• Answer sheets must be returned to the Ze Committee ZeMC office within 2.9 1. DO NOT OPEN THIS BOOKLET UNTIL YOU TELL YOU TO BEGIN.
seconds of the competition administration. Use an overnight or 2-day shipping 2. This is a 25 question multiple choice contest. For each question, only one answer choice
service, with a tracking number, to guarantee the timely arrival of these answer is correct.
sheets. If you wish for all of the answer sheets to get thrown in an incinerator,
USPS overnight is strongly recommended. 3. Submit your answers by PMing them through AoPS to “ihatemath123”, or DMing them
through Discord to “imagine dragon#3311”. If you use Discord, please specify your AoPS
• The publication, reproduction or communication of the problems or solutions of username.
You may format your answers in any way, as long as it is clear which problem each answer
this competition during the period when students are eligible to participate se-
corresponds to.
riously jeopardizes the integrity of the results. Dissemination via phone, email, If you wish to remain anonymous on the leaderboard, or wish to remain anonymous if
friends (if you have them), or digital media of any type during this period is a your score is below a certain threshold, make sure to specify this in your message.
violation of competition rules. DO NOT edit your message; you may be considered for cheating.
4. You should receive a response with your score and distribution within 24 hours, in addition
to a link with access to a private discussion forum.
The ZeMC competition series is made possible by the contributions of the following 5. SCORING: You will receive 1 point for each correct answer, 0 points for each problem
problem-writers and test-solvers: left unanswered, and 0 points for each incorrect answer.
6. Only blank scratch paper, rulers, and erasers are allowed as aids. Prohibited materials
Anchovy, asbodke, bissue, contactbibliophile, Geometry285, iamhungry, include calculators, smartwatches, phones, computing devices, compasses, protractors,
and graph paper. No problems on the contest will require the use of a calculator.
ihatemath123, Jiseop55406, kante314, Lasitha_Jayasinghe, mahaler,
Olympushero, peace09, P_Groudon, raagavbala, RithwikGupta, 7. Figures are not necessarily drawn to scale.
Significant, smileymittens and themathboi101. 8. You will have 40 minutes to complete the contest once you tell you to begin.

Thank you for taking our mock AMC!

The Ze Committee ZeMC Office reserves the right to disqualify scores from an individual if it determines
that the rules or the nonexistent required security procedures were not followed.

The publication, reproduction, or communication of the problems or solutions of this exam during
the period when students are eligible to participate seriously jeopardizes the integrity of the results.
Dissemination via phone, email, or digital media of any type during this period is a violation of the
competition rules.
 2022 ZeMC 8

1. Shelly cuts a square sheet of paper into triangular pieces, shown below. How
many pieces does she cut the paper into?

(A) 33 (B) 34 (C) 35 (D) 36 (E) 37

2. Ten darts are thrown at a dartboard. The darts land in the following marked
locations:
2
5
10

The point value of each region is shown. What is the average score earned from
all ten darts?

(A) 4.0 (B) 4.2 (C) 4.4 (D) 4.6 (E) 4.8

3. Luke glues two rectangles along one side, forming a square. If one of the rect-
angles has a width of 7 inches and a height of 5 inches, what is the perimeter,
in inches, of the other rectangle?

(A) 16 (B) 18 (C) 20 (D) 22 (E) 24

4. Asha is three times as old as Beth and Beth is twice as old as Chloe. If Asha
is 24 years older than Beth, how many years older is Beth than Chloe?

(A) 4 (B) 6 (C) 8 (D) 12 (E) 16

 2022 ZeMC 8

5. What is the product of the mixed fractions shown below?

1 2 3
1 2 3
2 3 4

(A) 12 (B) 15 (C) 16 (D) 18 (E) 20

6. The “?” in the equation below represents a single nonzero digit. What is that
digit?

508 × ?97 = 303,276

(A) 3 (B) 4 (C) 5 (D) 6 (E) 7

7. The table below shows the sunset and sunrise times for the longest and shortest
days of 2023 in Honolulu. How many minutes longer is the longest day than
the shortest day?

Sunrise Sunset
June 21, 2023 5:50 AM 7:16 PM
Dec 21, 2023 7:04 AM 5:55 PM

(A) 135 (B) 145 (C) 155 (D) 165 (E) 175

8. Anthony has 20 quarters and Bill has some dimes. The value of Anthony’s
quarters is twice as much as the value of Bill’s dimes. How many coins does
Bill have?

(A) 15 (B) 16 (C) 20 (D) 24 (E) 25

9. If a and b are positive integers such that (a + 1) × (b + 1) = 77, what is

(a − 1) × (b − 1)?

(A) 27 (B) 35 (C) 39 (D) 45 (E) 49

10. Jake and Jessica each pick a real number. The average of 20 and Jake’s number
is Jessica’s number. The average of 23 and Jessica’s number is Jake’s number.
How much larger is Jake’s number than Jessica’s number?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

 2022 ZeMC 8

11. Enya rolls a fair six-sided die four times, rolling a different number each time.
She notices that the product of the first two rolls is equal to the product of the
last two rolls. If the sum of the first two rolls is 8, what is the sum of the last
two rolls?

(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

12. A tax on candy bars is raised from 2% to 20%. A candy store offers a k %
discount on candy bars so that they cost as much as they did before the tax
raise. What is k ?

(A) 10 (B) 12 (C) 15 (D) 18 (E) 20

13. In the figure below, a unit square is surrounded by four, larger, congruent
squares, such that each of the larger squares partially shares an edge with the
unit square and two of the other larger squares. If the resulting figure has a
perimeter of 48, what is its area?

(A) 120 (B) 122 (C) 124 (D) 126 (E) 128

14. A farmer is milking and feeding 40 cows. Whenever a cow has been both fed
and milked, she receives a ribbon. After the farmer milks 40% and feeding 75%
of the cows, there are 11 cows with ribbons. How many cows have not been
fed or milked?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

 2022 ZeMC 8 2022 ZeMC 8

24. There are three clubs at a school: the math club, the physics club and the 15. Landon wrote an expression, shown below, using the four standard opera-
biology club. The following information is known: tions (addition, subtraction, multiplication, division) and no parentheses. His
brother erased all of the operations. However, Landon remembers that the
• Among the fifteen students in the math club, seven of them participate in expression was equal to an odd single-digit number. What is this number?
at least one other club.
4 6 2 8=?
• Among the twelve students in the physics club, three of them participate in
at least one other club.
(A) 1 (B) 3 (C) 5 (D) 7 (E) 9
• Among the ten students in the biology club, nine participate in at least one
other club.
16. Jason took ten quizzes in one semester of a class. The scores he earned are
How many students participate in at least one of these clubs? below, listed in the order he took the quizzes:
(A) 23 (B) 24 (C) 25 (D) 26 (E) 27 87, 90, 83, 95, 96, 92, 75, 89, 99, 91.
After receiving which grade was Jason’s average quiz score the highest?
25. In the diagram below, a square is inscribed in a circular sector with a measure
of 60◦ . If the area of the square is 36 and the area of the circular sector is Aπ , (A) 91 (B) 92 (C) 95 (D) 96 (E) 99
what is A?
17. Six students were assigned a summer reading list, consisting of five books,
titled A, B, C, D and E. The school librarian made a chart of which books
each student read. A corner of the chart was accidentally ripped off.

A B C D E
Ray
Victor
√ √ √ Charlie
(A) 12 + 6√2 (B) 18 + 2√ 2 (C) 16 + 2 6
(D) 14 + 4 3 (E) 18 + 2 3 Adi
Osvaldo

Alexandre

However, the librarian recalls the following information:

• More students read book E than any other book.
• Victor read more books than any other student.
• Ray read less books than any other student.
If c, d and e are the number of students that read books C , D and E , respec-
tively, what is c × d × e?

(A) 20 (B) 30 (C) 40 (D) 50 (E) 60

 2022 ZeMC 8 2022 ZeMC 8

18. In isosceles triangle ABC with AB = AC , let D be the unique point on side 21. Al, Bal, Cal and Dale each have a spinner, shown below. If each of them spin
AC for which AD = BD. If ∠BDC = 108◦ , what is the measure of ∠BCD ? their spinner twice and sum the two numbers they get, who is the most likely
to get a sum of 6?
A Al Bal
3 2 1 4
4 5 5 1

Cal Dale
3 2 3 2
2 4 1 3
D

108◦ (A) Al (B) Cal (C) Dale (D) Al & Bal are equally likely
(E) Bal & Dale are equally likely
B C
22. Forrest takes a two digit integer, finds the sum of its digits, and adds 36 to the
◦ ◦ ◦ ◦ ◦ result. When he reverses the digits of this sum, he gets the same number that
(A) 54 (B) 57 (C) 60 (D) 63 (E) 66
he started with. What is the sum of all possible numbers that Forrest could
have started with?
19. Mr. Bacon leaves his house to go to work at the same time every day. When he
drives to work at 60 miles per hour, he arrives at 8:29 AM. When he drives to (A) 405 (B) 444 (C) 445 (D) 486 (E) 495
work at 50 miles per hour, he arrives at 8:32 AM. What time does Mr. Bacon
leave his house every day? 23. A rectangular window is constructed out of four rectangular panes, whose areas
are shown in the diagram below. If the distance between the two marked points
(A) 8:11 AM (B) 8:12 AM (C) 8:13 AM
is 3, what is the perimeter of the entire window?
(D) 8:14 AM (E) 8:15 AM

20. Ann, Binh, Cara, Don, Erian, and Finn sat in a circle in that order, each
holding a number. Each person reported the sum of the two numbers held by 5
themselves and the person to the right of them:
19
Ann: 29. Binh: 32. Cara: 14. Don: 5. Erian: 26.

What number did Finn report?

(A) 28 (B) 29 (C) 30 (D) 31 (E) 32 7

5

(A) 22 (B) 23 (C) 24 (D) 25 (E) 26