AoPS Community 2023 AMC 8
AMC 8 2023
www.artofproblemsolving.com/community/c3198873
by peelybonehead, mannshah1211, pog, aops-g5-gethsemanea2, djmathman, hh99754539, r00tsOfUnity,
pi is 3.14, rrusczyk
1 What is the value of (8 × 4 + 2) − (8 + 4 × 2)?
(A) 0 (B) 6 (C) 10 (D) 18 (E) 24
2 A square piece of paper is folded twice into four equal quarters, as shown below, then cut along
the dashed line. When unfolded, the paper will match which of the following figures?
(A) (B) (C)
(D) (E)
3 Wind chill is a measure of how cold people feel when exposed to wind outside. A good estimate
for wind chill can be found using this calculation:
(wind chill) = (air temperature) − 0.7 × (wind speed),
where temperature is measured in degrees Fahrenheit (◦ F) and and the wind speed is measured
in miles per hour (mph). Suppose the air temperature is 36◦ F and the wind speed is 18 mph.
Which of the following is closest to the approximate wind chill?
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(A) 18 (B) 23 (C) 28 (D) 32 (E) 35
4 The numbers from 1 to 49 are arranged in a spiral pattern on a square grid, beginning at the
center. The first few numbers have been entered into the grid below. Consider the four numbers
that will appear in the shaded squares, on the same diagonal as the number 7. How many of
these four numbers are prime?
5 4 3
6 1 2
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
5 A lake contains 250 trout, along with a variety of other fish. When a marine biologist catches
and releases a sample of 180 fish from the lake, 30 are identified as trout. Assume that the ratio
of trout to the total number of fish is the same in both the sample and the lake. How many fish
are there in the lake?
(A) 1250 (B) 1500 (C) 1750 (D) 1800 (E) 2000
6 The digits 2, 0, 2, and 3 are placed in the expression below, one digit per box. What is the maxi-
mum possible value of the expression?
×
(A) 0 (B) 8 (C) 9 (D) 16 (E) 18
7 A rectangle, with sides parallel to the x−axis and y−axis, has opposite vertices located at (15, 3)
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and(16, 5). A line is drawn through points A(0, 0) and B(3, 1). Another line is drawn through
points C(0, 10) and D(2, 9). How many points on the rectangle lie on at least one of the two
lines?
C
10
D
9
8
7
6
5
4
3
2
B
1
A
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
8 Lola, Lolo, Tiya, and Tiyo participated in a ping pong tournament. Each player competed against
each of the other three players exactly twice. Shown below are the win-loss records for the
players. The numbers 1 and 0 represent a win or loss, respectively. For example, Lola won five
matches and lost the fourth match. What was Tiyo’s win-loss record?
Player Result
Lola 111011
Lolo 101010
Tiya 010100
Tiyo ??????
(A) 000101 (B) 001001 (C) 010000 (D) 010101 (E) 011000
9 Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the
base of the mountain as she skis along a trail. In total, how many seconds does she spend at
an elevation between 4 and 7 meters?
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16
14
12
Elevation (meters)
10
0
0 2 4 6 8 10 12 14 16
Time (seconds)
(A) 6 (B) 8 (C) 10 (D) 12 (E) 14
10 Harold made a plum pie to take on a picnic. He was able to eat only 14 of the pie, and he left
the rest for his friends. A moose came by and ate 31 of what Harold left behind. After that, a
porcupine ate 13 of what the moose left behind. How much of the original pie still remained after
the porcupine left?
1 1 1 1 5
(A) 12 (B) 6 (C) 4 (D) 3 (E) 12
11 NASA’s Perseverance Rover was launched on July 30, 2020. After traveling 292,526,838 miles, it
landed on Mars in Jezero Crater about 6.5 months later. Which of the following is closest to the
Rover’s average interplanetary speed in miles per hour?
(A) 6,000 (B) 12,000 (C) 60,000 (D) 120,000 (E) 600,000
12 The figure below shows a large white circle with a number of smaller white and shaded circles
in its interior. What fraction of the interior of the large white circle is shaded?
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1 11 1 19 5
(A) (B) (C) (D) (E)
4 36 3 36 9
13 Along the route of a bicycle race, 7 water stations are evenly spaced between the start and finish
lines, as shown in the figure below. There are also 2 repair stations evenly spaced between the
start and finish lines. The 3rd water station is located 2 miles after the 1st repair station. How
long is the race in miles?
Finish
Start
7
er
er
er
at
at
at
W
W
(A) 8 (B) 16 (C) 24 (D) 48 (E) 96
14 Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for
the postage, Nicolas would like to cover the package with a large number of stamps. Suppose
he has a collection of 5-cent, 10-cent, and 25-cent stamps, with exactly 20 of each type. What
is the greatest number of stamps Nicolas can use to make exactly $7.10 in postage?
(Note: The amount $7.10 corresponds to 7 dollars and 10 cents. One dollar is worth 100 cents.)
(A) 45 (B) 46 (C) 51 (D) 54 (E) 55
15 Viswam walks half a mile to get to school each day. His route consists of 10 city blocks of
equal length and he takes one minute to walk each block. Today, after walking 5 blocks, Viswam
discovers he has to take a detour, walking 3 blocks of equal length instead of one block to reach
the next corner. From the time he starts his detour what speed, in mph, must he walk in order
to get to school at his usual time?
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(A) 4.0 (B) 4.2 (C) 4.5 (D) 4.8 (E) 5.0
16 The letters P , Q, and R are entered in a 20 × 20 grid according to the pattern shown below. How
many P s, Qs, and Rs will appear in the completed table?
.. .. .. .. .. .
. . . . . ..
Q R P Q R ...
P Q R P Q ...
R P Q R P ...
Q R P Q R ...
P Q R P Q ...
(A) 132 Ps, 134 Qs, 134 Rs (B) 133 Ps, 133 Qs, 134 Rs (C) 133 Ps, 134 Qs, 133 Rs
(D) 134 Ps, 132 Qs, 134 Rs (E) 134 Ps, 133 Qs, 133 Rs
17 A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex.
Jun will make the regular octahedron shown on the right by folding the piece of paper shown
on the left. Which numbered face will end up to the right of Q?
Q ?
1 4 6
3 5 Q
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
18 Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump 5
pads to the right or 3 pads to the left. What is the fewest number of jumps Greta must make to
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reach the lily pad located 2023 pads to the right of her starting point?
(A) 405 (B) 407 (C) 409 (D) 411 (E) 413
19 An equilateral triangle is placed inside a larger equilateral triangle so that the region between
them can be divided into three congruent trapezoids, as shown below. The side length of the
inner triangle is 32 the side length of the larger triangle. What is the ratio of the area of one
trapezoid to the area of the inner triangle?
(A) 1 : 3 (B) 3 : 8 (C) 5 : 12 (D) 7 : 16 (E) 4 : 9
20 Two integers are inserted into the list 3, 3, 8, 11, 28 to double it’s range. The mode and median
remain unchanged. What is the maximum possible sum of two additional numbers?
(A) 56 (B) 57 (C) 58 (D) 60 (E) 61
21 Alina writes the numbers 1, 2, . . . , 9 on separate cards, one number per card. She wishes to
divide the cards into 3 groups of 3 cards so that the sum of the number in each group will be
the same. In how many ways can this be done?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
22 In a sequence of positive integers, each term after the second is the product of the previous two
terms. The sixth term in the sequence is 4000. What is the first term?
(A) 1 (B) 2 (C) 4 (D) 5 (E) 10
23 Each square in a 3 × 3 grid is randomly filled with one of the 4 gray-and-white tiles shown below
on the right.
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What is the probability that the tiling will contain a large gray diamond in one of the smaller 2 × 2
grids? Below is an example of one such tiling.
1 1 1 1 1
(A) 1024 (B) 256 (C) 64 (D) 16 (E) 4
24 Isosceles △ ABC has equal side lengths AB and BC. In the figure below, segments are drawn
parallel to AC so that the shaded portions of △ ABC have the same area. The heights of the
two unshaded portions are 11 and 5 units, respectively. What is the height of h of △ ABC?
B B
11
h
A C A C
(A) 14.6 (B) 14.8 (C) 15 (D) 15.2 (E) 15.4
25 Fifteen integers a1 , a2 , a3 , . . . , a15 are arranged in order on a number line. The integers are equally
spaced and have the property that
1 ≤ a1 ≤ 10, 13 ≤ a2 ≤ 20, 241 ≤ a15 ≤ 250.
What is the sum of digits of a14 ?
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(A) 8 (B) 9 (C) 10 (D) 11 (E) 12
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right © Mathematical Association of America (http://maa.org).
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