MAA American Mathematics Competitions

76th Annual

AMC 12 A
Wednesday, November 6, 2024

INSTRUCTIONS
1. DO NOT TURN TO THE NEXT PAGE UNTIL YOUR COMPETITION MANAGER TELLS YOU TO BEGIN.
2. This is a 25-question multiple-choice competition. For each question, only one answer choice is correct.
3. Mark your answer to each problem on the answer sheet with a #2 pencil. Check blackened answers for accuracy
and erase errors completely. Only answers that are properly marked on the answer sheet will be scored.
4. SCORING: You will receive 6 points for each correct answer, 1.5 points for each problem left unanswered, and 0
points for each incorrect answer.
5. Only blank scratch paper, rulers, compasses, and erasers are allowed as aids. No calculators, smartwatches, phones,
or computing devices are allowed. No problems on the competition will require the use of a calculator.
6. Figures are not necessarily drawn to scale.
7. You will have 75 minutes to complete the competition once your competition manager tells you to begin.
8. When you finish with the competition, please follow the directions of your competitions manager.

The problems and solutions for this AMC 12 A were prepared

by the MAA AMC 10/12 Editorial Board under the direction of
Gary Gordon and Carl Yerger, co-Editors-in-Chief.

The MAA AMC office reserves the right to disqualify scores from a school if it determines that the rules or the required security
procedures were not followed.
The publication, reproduction, or communication of the problems or solutions of this competition 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.
Students who score well on this AMC 12 will be invited to take the 43rd annual American Invitational Mathematics Examination (AIME)
on Wednesday, February 5, 2025, or Wednesday, February 12, 2025. More details about the AIME can be found at maa.org/AMC.

© 2024 Mathematical Association of America

2024 AMC 12 A Problems 2

Problem 1:
What is the value of 101 · 9,901 − 99 · 10,101 ?
(A) 2 (B) 20 (C) 21 (D) 200 (E) 2020

Problem 2:
A model used to estimate the time it will take to hike to the top of a mountain on a trail is of the
form T = aL + bG, where a and b are constants, T is the time in minutes, L is the length of the
trail in miles, and G is the altitude gain in feet. The model estimates that it will take 69 minutes
to hike to the top if a trail is 1.5 miles long and ascends 800 feet, as well as if a trail is 1.2 miles
long and ascends 1100 feet. How many minutes does the model estimate it will take to hike to the
top if the trail is 4.2 miles long and ascends 4000 feet?
(A) 240 (B) 246 (C) 252 (D) 258 (E) 264

Problem 3:
The number 2024 is written as the sum of not necessarily distinct two-digit numbers. What is the
least number of two-digit numbers needed to write this sum?
(A) 20 (B) 21 (C) 22 (D) 23 (E) 24

Problem 4:
What is the least value of n such that n! is a multiple of 2024 ?
(A) 11 (B) 21 (C) 22 (D) 23 (E) 253

Problem 5:
The product of three integers is 60. What is the least possible positive sum of the three integers?
(A) 2 (B) 3 (C) 5 (D) 6 (E) 13

Problem 6:
√
In △ABC, ∠ABC = 90◦ and BA = BC = 2. Points P1 , P2 , . . . , P2024 lie on hypotenuse AC so
that AP1 = P1 P2 = P2 P3 = · · · = P2023 P2024 = P2024 C. What is the length of the vector sum
−−→ −−→ −−→ −−−−→
BP1 + BP2 + BP3 + · · · + BP2024 ?

(A) 1011 (B) 1012 (C) 2023 (D) 2024 (E) 2025

Problem 7:
A data set containing 20 numbers, some of which are 6, has mean 45. When all the 6s are removed,
the data set has mean 66. How many 6s were in the original data set?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

Problem 8:
Let α be the radian measure of the smallest angle in a 3 – 4 – 5 right triangle. Let β be the radian
measure of the smallest angle in a 7 – 24 – 25 right triangle. In terms of α, what is β ?
α π π α
(A) (B) α − (C) − 2α (D) (E) π − 4α
3 8 2 2
2024 AMC 12 A Problems 3

Problem 9:
How many angles θ with 0 ≤ θ ≤ 2π satisfy log (sin(3θ)) + log (cos(2θ)) = 0 ?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Problem 10:
Let M be the greatest integer such that both M + 1213 and M + 3773 are perfect squares. What
is the units digit of M ?
(A) 1 (B) 2 (C) 3 (D) 6 (E) 8

Problem 11:
The first three terms of a geometric sequence are the integers a, 720, and b, where a < 720 < b.
What is the sum of the digits of the least possible value of b ?
(A) 9 (B) 12 (C) 16 (D) 18 (E) 21

Problem 12:
The graph of y = ex+1 + e−x − 2 has an axis of symmetry. What is the reflection of the point
1


−1, 2 over this axis?
       
3 1 1 1
(A) −1, − (B) (−1, 0) (C) −1, (D) 0, (E) 3,
2 2 2 2
Problem 13:
There are exactly K positive integers b with 5 ≤ b ≤ 2024 such that the base-b integer 2024b is
divisible by 16 (where 16 is in base ten). What is the sum of the digits of K ?
(A) 16 (B) 17 (C) 18 (D) 20 (E) 21

Problem 14:
A set of 12 tokens—3 red, 2 white, 1 blue, and 6 black—is to be distributed at random to 3 game
players, 4 tokens per player. The probability that some player gets all the red tokens, another
player gets all the white tokens, and the remaining player gets the blue token can be written as m
n,
where m and n are relatively prime positive integers. What is m + n ?
(A) 387 (B) 388 (C) 389 (D) 390 (E) 391

Problem 15:
The numbers, in order, of each row and the numbers, in order, of each column of a 5 × 5 array of
integers form an arithmetic progression of length 5. The numbers in positions (5, 5), (2, 4), (4, 3),
and (3, 1) are 0, 48, 16, and 12, respectively. What number is in position (1, 2) ?
 
. ? . . .
 . . . 48 . 
 
12 . . . . 
 
 . . 16 . . 
. . . . 0

(A) 19 (B) 24 (C) 29 (D) 34 (E) 39

2024 AMC 12 A Problems 4

Problem 16:
The roots of x3 + 2x2 − x + 3 are p, q, and r. What is the value of

p2 + 4 q 2 + 4 r2 + 4 ?

(A) 64 (B) 75 (C) 100 (D) 125 (E) 144

Problem 17:
√
On top of a rectangular card with sides of length 1 and 2 + 3, an identical card is placed so that
two of their diagonals line up, as shown (AC, in this case).

B A

Continue the process, adding a third card to the second, and so on, lining up successive diagonals
after rotating clockwise. In total, how many cards must be used until a vertex of a new card lands
exactly on the vertex labeled B in the figure?
(A) 6 (B) 8 (C) 10 (D) 12 (E) No new vertex will land on B.

Problem 18:
Cyclic quadrilateral ABCD has lengths BC = CD = 3 and DA = 5 with ∠CDA = 120◦ . What is
the length of the shorter diagonal of ABCD ?
31 33 39 41
(A) (B) (C) 5 (D) (E)
7 7 7 7
Problem 19:
Integers a, b, and c satisfy ab + c = 100, bc + a = 87, and ca + b = 60. What is ab + bc + ca ?
(A) 212 (B) 247 (C) 258 (D) 276 (E) 284
2024 AMC 12 A Problems 5

Problem 20:
The figure below shows a dotted grid 8 cells wide and 3 cells tall consisting of 1′′ × 1′′ squares. Carl
places 1-inch toothpicks along some of the sides of the squares to create a closed loop that does not
intersect itself. The numbers in the cells indicate the number of sides of that square that are to be
covered by toothpicks, and any number of toothpicks are allowed if no number is written. In how
many ways can Carl place the toothpicks?

1 1 1 1 1 1 1 1

(A) 130 (B) 144 (C) 146 (D) 162 (E) 196

Problem 21:
Points P and Q are chosen uniformly and independently at random on sides AB and AC, respec-
tively, of equilateral triangle △ABC. Which of the following intervals contains the probability that
the area of △AP Q is less than half the area of △ABC ?
         
3 1 1 2 2 3 3 7 7
(A) , (B) , (C) , (D) , (E) ,1
8 2 2 3 3 4 4 8 8
Problem 22:
Suppose that a1 = 2 and the sequence (an ) satisfies the recurrence relation
an − 1 an−1 + 1
=
n−1 (n − 1) + 1

for all n ≥ 2. What is the greatest integer less than or equal to

100
X
a2n ?
n=1

(A) 338,550 (B) 338,551 (C) 338,552 (D) 338,553 (E) 338,554

Problem 23:
What is the value of
π 3π π 5π 3π 7π 5π 7π
tan2 · tan2 + tan2 · tan2 + tan2 · tan2 + tan2 · tan2 ?
16 16 16 16 16 16 16 16

(A) 28 (B) 68 (C) 70 (D) 72 (E) 84

Problem 24:
A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the
least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
√ √ √ √
(A) 3 (B) 3 15 (C) 15 (D) 15 7 (E) 24 6
 2024 AMC 12 A Problems 6

Problem 25:
A graph is symmetric about a line if the graph remains unchanged after reflection in that line. For
how many quadruples of integers (a, b, c, d), where |a|, |b|, |c|, |d| ≤ 5 and c and d are not both 0, is
the graph of
ax + b
y=
cx + d
symmetric about the line y = x ?
(A) 1282 (B) 1292 (C) 1310 (D) 1320 (E) 1330

Results will be available online at https://asiamathsalliance.com/honor-rollresults from

12th Jan 2025 or earlier.

School candidates will be able to obtain their results and details of their invitation for the American
Invitational Mathematics Exam (AIME-II)via their school

American Invitational Mathematics Exam-II(AIME-II) 2025

 Only the AIME-II will be offered to candidates outside of the U.S/Canada from 2025
 The AIME is a 3-hour 15 question paper whereby answers are integers between 0-999
 Coaching sessions for AIME-II will commenced from 11th
 2024-2025 Forthcoming Math/Science Competitions
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Students aged 17.5

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Students aged 17.5

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American Invitational
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