1 Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B. Alicia bikes towards B at
18 miles per hour. Leaving at the same time, Beth bikes toward A at 12 miles per hour. How many
miles from City A will they be when they meet? (A) 20 (B) 24 (C) 25 (D) 26 (E) 27
2 The weight of 1
3
of a large pizza together with 3
1
2
cups of orange slices is the same as the weight
of 3
4
of a large pizza together with 1
2
cup of orange slices. A cup of orange slices weighs 1
4
of a
pound. What is the weight, in pounds, of a large pizza?
(A) 1
4
5
(B) 2 (C) 2
2
5
(D) 3 (E) 3
3
5
3 How many positive perfect squares less than 2023 are divisible by 5?
(A) 8 (B) 9 (C) 10 (D) 11 (E) 12
4 How many digits are in the base-ten representation of 8
5
· 5
10
· 155
?
(A) 14 (B) 15 (C) 16 (D) 17 (E) 18
5 Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls.
What is the probability that at some point, her running total will equal 3?
(A) 2
9
(B) 49
216 (C) 25
108 (D) 17
72 (E) 13
54
6 Points A and B lie on the graph of y = log2 x. The midpoint of AB is (6, 2). What is the positive
difference between the x-coordinates of A and B?
(A) 2
√
11 (B) 4
√
3 (C) 8 (D) 4
√
5 (E) 9
7 A digital display shows the current date as an 8-digit integer consisting of a 4-digit year, followed
by a 2-digit month, followed by a 2-digit date within the month. For example, Arbor Day this year
is displayed as 20230428. For how many dates in 2023 will each digit appear an even number
of times in the 8-digital display for that date?
 (A) 5 (B) 6 (C) 7 (D) 8 (E) 9
8 Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores
an 11 on the next quiz, her mean will increase by 1. If she scores an 11 on each of the next
three quizzes, her mean will increase by 2. What is the mean of her quiz scores currently?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8
9 A square of area 2 is inscribed in a square of area 3, creating four congruent triangles, as shown
below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
(A) 1
5
(B) 1
4
(C) 2 −
√
3 (D) √
3 −
√
2 (E) √
2 − 1
10 Positive real numbers x and y satisfy y
3 = x
2 and (y − x)
2 = 4y
2
. What is x + y?
(A) 12 (B) 18 (C) 24 (D) 36 (E) 42
11 What is the degree measure of the acute angle formed by lines with slopes 2 and 1
3
?
(A) 30 (B) 37.5 (C) 45 (D) 52.5 (E) 60
12 What is the value of
2
3 − 1
2 + 43 − 3
3 + 63 − 5
3 + · · · + 183 − 173
?
(A) 2023 (B) 2679 (C) 2941 (D) 3159 (E) 3235
13 In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was 40% more than the number of games won by righthanded players. (There were no ties and no ambidextrous players.) What is the total number of
games played?
(A) 15 (B) 36 (C) 45 (D) 48 (E) 66
14 How many complex numbers satisfy the equation z
5 = z, where z is the conjugate of the complex number z?
(A) 2 (B) 3 (C) 5 (D) 6 (E) 7
15 Usain is walking for exercise by zigzagging across a 100-meter by 30-meter rectangular field,
beginning at point A and ending on the segment BC. He wants to increase the distance walked
by zigzagging as shown in the figure below (AP QRS). What angle θ = ∠P AB = ∠QP C =
∠RQB = · · · will produce in a length that is 120 meters? (Do not assume the zigzag path has
exactly four segments as shown; there could be more or fewer.) (A) arccos 5
6
(B) arccos 4
5
(C) arccos 3
10 (D) arcsin 4
5
(E) arcsin 5
6
16 Consider the set of complex numbers z satisfying |1 + z + z
2
| = 4. The maximum value of the
imaginary part of z can be written in the form
√
m
n
, where m and n are relatively prime positive
integers. What is m + n?
(A) 20 (B) 21 (C) 22 (D) 23 (E) 24
17 Flora the frog starts at 0 on the number line and makes a sequence of jumps to the right. In
any one jump, independent of previous jumps, Flora leaps a positive integer distance m with
probability 1
2m . What is the probability that Flora will eventually land at 10?
(A) 5
512 (B) 45
1024 (C) 127
1024 (D) 511
1024 (E) 1
2
18 Circle C1 and C2 each have radius 1, and the distance between their centers is 1
2
. Circle C3 is the
largest circle internally tangent to both C1 and C2. Circle C4 is internally tangent to both C1 and
C2 and externally tangent to C3. What is the radius of C4?
(A) 1
14 (B) 1
12 (C) 1
10 (D) 3
28 (E) 1
9
19 What is the product of all the solutions to the equation
log7x 2023 · log289x 2023 = log2023x 2023?
(A) (log2023 7·log2023 289)2
(B) log2023 7·log2023 289 (C) 1 (D) log7 2023·log289 2023 (E) (log7 20log289 2023)2
20 Rows 1, 2, 3, 4, and 5 of a triangular array of integers are shown below:
1
1 1
1 3 1
1 5 5 1
1 7 11 7 1
Each row after the first row is formed by placing a 1 at each end of the row, and each interior
entry is 1 greater than the sum of the two numbers diagonally above it in the previous row. What
is the units digit of the sum of the 2023 numbers in the 2023rd row?
(A) 1 (B) 3 (C) 5 (D) 7 (E) 9
21 If A and B are vertices of a polyhedron, define the distance d(A, B) to be the minimum number
of edges of the polyhedron one must traverse in order to connect A and B. For example, if AB
© 2023 AoPS Incorporated 4
AoPS Community 2023 AMC 12/AHSME
is an edge of the polyhedron, then d(A, B) = 1, but if AC and CB are edges and AB is not
an edge, then d(A, B) = 2. Let Q, R, and S be randomly chosen distinct vertices of a regular
icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability
that d(Q, R) > d(R, S)?
(A) 7
22 (B) 1
3
(C) 3
8
(D) 5
12 (E) 1
2
22 Let f be the unique function defined on the positive integers such that
X
d|n
d · f
n
d

= 1
for all positive integers n, where the sum is taken over all positive divisors of n. What is f(2023)?
(A) − 1536 (B) 96 (C) 108 (D) 116 (E) 144
23 How many ordered pairs of positive real numbers (a, b) satisfy the equation
(1 + 2a)(2 + 2b)(2a + b) = 32ab?
(A) 0 (B) 1 (C) 2 (D) 3 (E) an infinite number
24 Let K be the number of sequences A1, A2, . . . , An such that n is a positive integer less than or
equal to 10, each Ai
is a subset of {1, 2, 3, . . . , 10}, and Ai−1 is a subset of Ai for each i between
2 and n, inclusive. For example, {}, {5, 7}, {2, 5, 7}, {2, 5, 7}, {2, 5, 6, 7, 9} is one such sequence,
with n = 5. What is the remainder when K is divided by 10?
(A) 1 (B) 3 (C) 5 (D) 7 (E) 9
25 There is a unique sequence of integers a1, a2, · · · a2023 such that
tan 2023x =
a1 tan x + a3 tan3 x + a5 tan5 x + · · · + a2023 tan2023 x
1 + a2 tan2 x + a4 tan4 x · · · + a2022 tan2022 x
whenever tan 2023x is defined. What is a2023?
(A) − 2023 (B) − 2022 (C) − 1 (D) 1 (E) 2023