2022

XYZMC
XYZMC 10 XYZ Mathematics Competition
FIRST ANNUAL
REMAINS OPEN UNTIL THE DUE DATE
XYZMC 10
**Administration On An Earlier Date Is Not Even Possible**
relaXing Yields Zen! 10

1. All information (Rules and Instructions) needed to administer this exam is contained in the TEACH- INSTRUCTIONS
ERS’ MANUAL. PLEASE READ THE MANUAL BEFORE In ur mom.
1. DO NOT OPEN THIS BOOKLET UNTIL THE TIMER STARTS.
2. Your PRINCIPAL or VICE-PRINCIPAL must verify on the AMC 10 CERTIFICATION FORM
(found in the Teachers’ Manual) that you followed all rules associated with the conduct of the exam. 2. This is a twenty-five question multiple choice test. Each question is followed by answers marked A,
B, C, D and E. Only one of these is correct.
3. The Answer Forms must be mailed by trackable mail to the AMC office no later than 24 hours
following the exam. 3. Mark your answer clearly, edits in submission will not be accepted.

4. The publication, reproduction or communication of the problems or solutions for this contest during 4. SCORING: You will receive 1 point for each correct answer, 0 points for each problem left unan-
the period when students are eligible to participate seriously jeopardizes the integrity of the results. swered, and 0 points for each incorrect answer.
Dissemination at any time via copier, telephone, email, internet, or media of any type is a violation
of the competition rules. 5. No aids are permitted other than scratch paper, graph paper, rulers, compass, protractors, and
erasers. No calculators, smartwatches, or computing devices are allowed. No problems on the test
will require the use of a calculator.
6. Figures are not necessarily drawn to scale.
7. Before beginning the test, your proctor will ask you to record certain information on the answer
The relaXing Yields Zen! are brought to you by:
form. This includes zipcode so we can track you down.
mannshah1211, 8. When your proctor gives the signal, begin working on the problems. You will have 40 minutes to
and the XYZ AMC Team complete the test.
9. When you finish the exam, sign your name in the space provided on the Answer Form to get ur
name revealed.

The XYZ AMC Team (XYZMC) reserves the right to re-examine students before deciding whether to grant official status
to their scores.
2022 XYZMC 10

22 + 11 22 + 22 22 + 33
1. What is the value of + + ?
22 + 0 · 0 22 + 0 · 0 22 + 0 · 0
(A) 10 (B) 10.5 (C) 11 (D) 11.5 (E) 12

2. Dan went to watch a 3 hours and 45 minute movie. Halfway through the movie, he
goes to get popcorn for 18 minutes. How much percent of the movie was he able to
watch?
(A) 80 (B) 84 (C) 88 (D) 92 (E) 96

3. Amy has an equal number of quarters, dimes, nickels, and pennies. If she has $8.61,
how many coins does she have?
(A) 80 (B) 84 (C) 88 (D) 92 (E) 96

4. Chad has multiple $10 dollar bills. He can buy at most 46 socks worth 3 dollars each.
At most, how many shoes can he buy if each shoe is worth 12 dollars?
(A) 10 (B) 11 (C) 12 (D) 13 (E) 14

5. There exists a two digit number, xy such that 6xy9 is divisible by 59. What is x + y?
(A) 8 (B) 9 (C) 10 (D) 11 (E) 12

6. A train passes through a 2000 meter tunnel and a 1200 meter tunnel completely inside
for 23 seconds and 13 seconds, respectively. If the train passes through each tunnel
at the same speed, how long, in meters, is the train?
(A) 100 (B) 120 (C) 140 (D) 160 (E) 180

7. A move is defined as one of the following

22. An equilateral triangle with side length 12 is placed in space. Two points, X and Y 8. Pete the Pirate has trouble understanding the four directions, and knows nothing
are so placed such that the distances from both the points to each of the vertices are other than that North is opposite to South, and East is opposite to West. Without
2
12, 12, and 6 respectively. Find the sum of all distinct values of XY . a compass in hand, he has to follow a map with the following written instructions in
order to uncover a treasure:
(A) 426 (B) 474 (C) 606 (D) 654 (E) 786
1. Walk north 5 miles 3. Walk south 5 miles

23. Let ABCD be a trapezoid with AB ∥ DC and ∠C and ∠D acute. Suppose ABCD 2. Walk east 6 miles 4. Walk west 3 miles
has an inscribed circle with center O and radius 2. If AD = 5 and the area of △BOD Using what he knows, he will randomly assign North, East, South, and West to the
is 2.5, what is the length of BC? four actual directions. Assuming that Pete strictly follows the map otherwise, what
25 13 9 14 is the probability that he will obtain the treasure?
(A) 4 (B) (C) (D) (E)
6 3 2 3 1 1 1 1 1
(A) (B) (C) (D) (E)
12 8 6 4 2
24. Alice and Bob are playing a game. Bob chooses a positive integer, k and Alice chooses
two relatively prime positive integers, p and q such that p < q < k7 . If k cannot be m
9. Consider the fraction where m is a positive integer. Suppose that it is simplified
written as ap + bq for non-negative integers a and b, Alice wins, otherwise Bob wins. 40
p
What is the remainder when the maximum possible value of k where Bob wins is to where p and q are relatively prime positive integers. If p + q = 59, what is the
q
divided by 5 if Alice and Bob both play accordingly? sum of all possible values of p?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4 (A) 374 (B) 382 (C) 390 (D) 394 (E) 402

25. Three non-collinear lattice points are chosen from a 5 by 5 array such that there 10. Two smaller regular hexagons, are put inside a larger regular hexagon. If the smallest
exists a parabola through the points that either faces directly upwards or directly hexagon has an area of 24, what is the area of the largest hexagon?
downwards. How many ways exist such that the vertex of the parabola goes through
another lattice point, different than the three originally selected points?
(A) 1950 (B) 1956 (C) 1960 (D) 1972 (E) 1980

(A) 360 (B) 372 (C) 384 (D) 392 (E) 400
√
11. How many ordered pairs (a, b) of positive integers satisfy the equation ab ab =
216000?
(A) 30 (B) 36 (C) 45 (D) 48 (E) 56
 2022 XYZMC 10 2022 XYZMC 10

12. Caleb is playing with 9 identical unit cubes. He lines up multiple stacks of cubes in 17. Eve chooses a group of 6, (not always distinct) integers from the the set {1, 2, 3, . . . , 12}.
a row. Define the altitude of his structure as the distance between the highest point For how many such groups would the unique mode of the numbers be 6?
and the ground. How many such structures can he build with an altitude of 3?
(A) 564 (B) 617 (C) 672 (D) 683 (E) 694
(A) 87 (B) 88 (C) 90 (D) 93 (E) 94
18. Let the interval of all values of k be (m, n) such that there exist exactly 6 distinct
13. Consider a rectangle ABCD with AB = 6 and BC = 8. Let M and N be the solutions for x in
midpoints of AB and BC respectively. What is the length of the perpendicular
drawn from D to M N ? |300 − |200 − |x + 100||| = k,

(A) 4.8 (B) 6.4 (C) 7 (D) 7.2 (E) 8.4 for some k. What is |n − m|?
(A) 100 (B) 150 (C) 200 (D) 250 (E) 300
14. Call a number satisfactory if all its even digits appear in non-decreasing order from
left to right and all of its odd digits appear in non-decreasing order from left to right.
How many 3 digit satisfactory numbers exist? 19. Ed has two Us and two Zs, initially arranged in the order UZUZ. Every second, he
randomly chooses one of the two Us and one of the two Zs and switches their place-
(A) 400 (B) 410 (C) 420 (D) 430 (E) 440 ments. What is the expected number of seconds until Ed forms the string ZUZU?
9 11
15. Let a, b, c be the sides of a triangle with semi perimeter s. If s > 2b, then (A) 4 (B) (C) 5 (D) (E) 6
2 2
(A) (s − 3c) must be non positive.
(B) (s − 3c) must be strictly positive. 20. A company has 2023 employees. They all have ID’s from 1 to 2023, and individual
(C) (s − 3c) must be strictly negative. offices in a row in increasing order. On any given day, Employee with ID 1 has a 12
(D) Information is not sufficient to conclude the sign of (s − 3c). sn
chance of staying overtime. For all n ≥ 2, Employee with ID n has a probability of
(E) Such a Triangle cannot exist. n
of staying overtime where sn represents the number of employees with ID smaller than
16. There exists a triangle with 2 parallel lines perpendicular to the hypotenuse and n staying overtime. What is the expected number of employees that stay overtime?
intersecting each of the legs, as well as a line parallel to the hypotenuse intersecting
both legs. This splits the plane into 6 bounded sections: 5 triangles of equal area, 1011 2023 1013
√ (A) 505 (B) (C) (D) 506 (E)
and a hexagon of area 6 + 32. What is the distance between the two parallel lines 2 4 2
perpendicular to the hypotenuse?
21. Let f (x) = (x − 2)(x − 3) and g(x) = 2(x + 5). Three random real numbers a, b, and
c are chosen, respectively, from the range (−10, 20). Which of the following is closest
to the probability that
f (a) + f (b) + f (c) > g(a) + g(b) + g(c)?

(A) 95% (B) 96% (C) 97% (D) 98% (E) 99%

√ √ √ √ √
(A) 2+1 (B) 2+2 (C) 2 2 + 1 (D) 3 2 (E) 2 2 + 2