MSI 2021 AMC10 Mock Exam

• Answers are on the last page

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Problem 1 If
1
2
× 13 × 14 × 51 + 32 × 34 × 3
5 m
1 =
× 23 × 52 n

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for two co-prime positive integers m and n, what is m + n?
(A) 25 (B) 33 (C) 37 (D) 45 (E) 49

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Problem 2 What is the number of digits of N = 167 × 34 × 525 ?
(A) 25 (B) 26 (C) 27 (D) 28 (E) 26
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Problem 3 [Australian Math Competition] Two machines move at constant speeds
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around a circle of circumference 600 cm, starting together from the same
point. If they travel in the same direction then they next meet after 20
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seconds, but if they travel in opposite directions then they next meet after
5 seconds. At what speed, in centimetres per second, is the faster one
travelling?
(A) 60 (B) 65 (C) 70 (D) 75 (E) 80
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Problem 4 [Canadian Math Competition] Juliana chooses three different numbers

from the set {−6, −4, −2, 0, 1, 3, 5, 7} and multiplies them together to obtain
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the integer n. What is the greatest possible value of n?

or

(A) 0 (B) 15 (C) 105 (D) 168 (E) 210

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Problem 5 What is the number of integers solutions (x, y, z) to

x2 y 2 z 2 = 36 ?

(A) 16 (B) 24 (C) 48 (D) 72 (E) 80

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 Problem 6 a < b < c < d are four positive integers and a + b + c + d = 100. What is
the largest possible value of a + b?
(A) 44 (B) 45 (C) 46 (D) 47 (E) 48

Problem 7 A construction team is building a tunnel. When 13 of the tunnel is finished,

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the construction team receives a new equipment which can improve the
working speed by 20%. Meanwhile, in order to maintain this new equipment,
each day’s working time is cut by 15 . It takes the construction team 185 days

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in total to finish the bridge. How many days do they need to finish the
project if the new equipment is not used and working schedule does not
change?
(A) 171 (B) 174 (C) 180 (D) 183 (E) 189

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Problem 8 For two real numbers x and y, define
x ⊗ y = x2 − y 2 .
What is the last two digits of
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5 ⊗ 6 − (6 ⊗ 7) − (7 ⊗ 8) − · · · − (99 ⊗ 100)?
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(A) 46 (B) 53 (C) 61 (D) 70 (E) 95
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Problem 9 If 8 is the smallest possible value of

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p(x) = (x − 1)2 + (x − 3)2 + m.

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for a give real number m, what is m?

(A) 2 (B) 4 (C) 5 (D) 6 (E) 8
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Problem 10 2021 people line up from left to right. At the beginning, each person is
assigned an integer number from left to right: 1, 2, 3, 4,..., 2021. Next,
people with odd label numbers are asked to leave the line. Others remain
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in the line and are reassigned an integer label number from left to right: 1,
2, 3, 4,..., 1010. Repeat this procedure until there is only one person left in
the line. If N is the number that is assigned to this person at the beginning,
what is the sum of digits of N ?
(A) 7 (B) 8 (C) 9 (D) 10 (E) 11

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2 Morning Star Institute, Irvine CA, USA
Problem 11 What is the area of ∆ABC whose three heights are 12, 15 and 20 respec-
tively?
(A) 90 (B) 100 (C) 120 (D) 130 (E) 150

Problem 12 [USMCA Challenger] The Fibonacci sequence F0 , F1 , · · · satisfies F0 = 0,

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F1 = 1, and Fn+2 = Fn+1 + Fn for all n ≥ 0. Compute the number of triples
(a, b, c) with 0 ≤ a < b < c ≤ 100 for which Fa , Fb , Fc is an arithmetic
progression.

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(A) 101 (B) 102 (C) 103 (D) 104 (E) 105

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Problem 13 [Hope Cup Math Competition, China] B and C are two points on

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circle O. A is point inside the circle. If ∠B = ∠A = 60◦ , AB = 8 and
BC = 12, what is the area of the circle?
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(A) 40π (B) 48π (C) 50π (D) 52π (E) 60π
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Problem 14 [Chinese High School Math Competition] Let

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A = {(x, y)| |x| + |y| = a}

and
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B = {(x, y)| |xy| + 1 = |x| + |y|}.

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Here a is a positive number. If A ∩ B is the collection of eight vertices of a

regular octagon, which of the following integer is closest to the sum of all
possible values of a?
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(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

Problem 15 Randomly choose two numbers x and y between 0 and 1. What is the
probability that none of these two numbers is the median of the following
set  
1 1 1
x, y, , , ?
2 4 8
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Morning Star Institute, Irvine CA, USA 3
 1 17 5 41 3
(A) 2
(B) 32
(C) 8
(D) 64
(E) 4

Problem 16 Three friends A, B and C play chess. Every two of them play once. For
each game, the winner gets 2 points, the loser gets 0 point. If it is a tie,
each player gets one point. What is the number of different arrangements

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of scores after 3 games? For example, the triple (A = 4, B = 1, C = 1)
represents the arrangement of scores where A gets 4 points, B gets 1 point
and C gets 1 point.

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(A) 15 (B) 19 (C) 22 (D) 24 (E) 27

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Problem 17 The straight line x4 + y3 = 1 intersects the ellipse x16 + y9 = 1 at two points

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A and B. Let P1 and P2 are two points on the ellipse such that the areas
of 4P1 AB and 4P2 AB are both 6. What is the area of the quadrilateral
ABP1 P2 ?
√
(A) 7 3 (B) 13
a (C) 14 rI (D) 6(1 +
√
2) (E) 44
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Problem 18 If c is the largest integer such that the polynomial

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P (x) = x6 − ax4 + bx2 + c

has 6 distinct integer roots that form an arithmetic sequence. What is a?
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(A) 27 (B) 35 (C) 40 (D) 53 (E) 61

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Problem 19 In number theory, a Smith number is a composite number for which, the
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sum of its digits is equal to the sum of the digits in its prime factorization.
For example, 58 and 378 are both smith number since
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58 = 2 × 29 and 5 + 8 = 2 + 2 + 9 = 13
and
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378 = 21 × 33 × 71 and 3 + 7 + 8 = 2 × 1 + 3 × 3 + 7 × 1.
Find the sum of all possible prime numbers that are less than 100 and 12p
is a smith number.
(A) 60 (B) 72 (C) 88 (D) 124 (E) 168

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4 Morning Star Institute, Irvine CA, USA
Problem 20 M is a circular cone frustum with height 18. Inside M , a sphere O1 with
radius 7 is tangent to the bottom and wall. A sphere O2 with radius 3 is
tangent to O1 and the wall and top of M . How many more spheres with
radius 3 can be put inside M and above O1 ? Find the largest possible
number.

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(A) 2 (B) 3
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(C) 4
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(D) 5 (E) 6
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Problem 21 Five men line up to board a bus that has 5 available seats. Each man has
a ticket with assigned seat. However, the first person to board has lost his
ticket and forgets his seat number. So he takes a random seat. After that,
each person takes the assigned seat if it is unoccupied, and one of unoccupied
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seats at random otherwise. What is the probability that the fourth person
to board gets to sit in his assigned seat?
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1 1 1 2 3
(A) 5
(B) 3
(C) 2
(D) 3
(E) 4
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Problem 22 Let a, b, and c be real numbers that satisfy

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ab + c + 5 = 0
bc + a + 5 = 0
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ca + b + 5 = 0.
Find the sum of all possible values of a2 + b − c.
(A)18 (B)28 (C)38 (D)48 (E)58

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Problem 23 [Hope Cup Math Competition, China] ABCD is a cyclic quadrilateral.
AB = 2, AD = CD = 4 and BC = 6. What is BD?

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√ √ 11
(A) 5 (B) 2 7 (C) 4 2 (D) 2
(E) 6

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Problem 24 [Ying Chun Cup Math Competition, China] Consider the sum of all
three-digit numbers whose remainders on division by 4, 9 and 25 sum up to
17. What is the remainder of that sum on division by 900?
(A) 111 (B) 212
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(C) 313 (D) 414 (E) 515

Problem 25 [Chinese High School Math Competition] There are 10 cards. Each
card has two different numbers from 1, 2, 3,4, 5 on it. No two cards have
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the same two numbers. Put these 10 cards into 5 boxes labeled 1, 2,3,4,5.
A card with numbers i and j on it can only be put in either box i or box j.
Let N be the number of ways we can place these 10 cards so that box 1 has
more cards than any other box. What is the sum of digits of N ?
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(A) 3 (B) 6 (C) 9 (D) 12 (E) 15

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 Answers:

1. E 2. D 3. D 4. D 5. D
6. D 7. C 8. B 9. D 10. A
11. E 12. A 13 B 14. D 15. D

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16. B 17. D 18. B 19. D 20 D
21. D 22. C 23. B 24. D 25. A

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• Solution Videos are available on our website:
http://www.morningstarinstitute.org/amc-mock-tests/
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