2024 AMC 12A Problems

2024 AMC 12A (Answer Key) Printable versions: Wiki • AoPS Resources • PDF
Instructions This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct. You will receive 6 points for each correct answer, 2.5 points for each problem left unanswered if the year is before 2006, 1.5 points for each problem left unanswered if the year is after 2006, and 0 points for each incorrect answer. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers (and calculators that are accepted for use on the test if before 2006. No problems on the test will require the use of a calculator). Figures are not necessarily drawn to scale. You will have 75 minutes working time to complete the test.
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Problem 1

What is the value of $9901 \cdot 101 - 99 \cdot 10101$ ?

$\textbf{(A)}~2\qquad\textbf{(B)}~20\qquad\textbf{(C)}~200\qquad\textbf{(D)}~202\qquad\textbf{(E)}~2020$

Solution

Problem 2

Define $\blacktriangledown(a) = \sqrt{a - 1}$ and $\blacktriangle(a) = \sqrt{a + 1}$ for all real numbers $a$ . What is the value of $\frac{\blacktriangledown(20 + \blacktriangle(2024))}{\blacktriangledown(\blacktriangle(24))}~?$

$\textbf{(A)}~ 1 \qquad \textbf{(B)}~ 2 \qquad \textbf{(C)}~ 4 \qquad \textbf{(D)}~ 8 \qquad \textbf{(E)}~ 16$

Solution

Problem 3

A square and an isosceles triangle are joined along an edge to form a pentagon $10$ inches tall and $22$ inches wide, as shown below. What is the perimeter of the pentagon, in inches?

$[asy] import graph; size(7cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ pen GGG = grey; draw((10, 0)--(0, 0)--(0, 10)--(10, 10)); draw((10, 0)--(10, 10), dashed); draw((10, 0)--(22, 5)--(10, 10)); draw((-1.5, 0)--(-1.5, 10), arrow = ArcArrow(SimpleHead), GGG); draw((-1.5, 10)--(-1.5, 0), arrow = ArcArrow(SimpleHead), GGG); draw((0, 11.5)--(22, 11.5), arrow = ArcArrow(SimpleHead), GGG); draw((22, 11.5)--(0, 11.5), arrow = ArcArrow(SimpleHead), GGG); label("$10$ in.", (-3.5, 5), GGG); label("$22$ in.", (11, 12.75), GGG); dot((0, 0)); dot((0, 10)); dot((10, 10)); dot((10, 0)); dot((22, 5)); [/asy]$

$\textbf{(A)}~54\qquad \textbf{(B)}~56 \qquad \textbf{(C)}~62 \qquad \textbf{(D)}~64 \qquad \textbf{(E)}~66$

Solution

Problem 4

A data set containing $20$ numbers, some of which are $6$ , has mean $45$ . When all the $6$ s are removed, the data set has mean $66$ . How many $6$ s were in the original data set?

$\textbf{(A)}~4\qquad\textbf{(B)}~5\qquad\textbf{(C)}~6\qquad\textbf{(D)}~7\qquad\textbf{(E)}~8$

Solution

Problem 5

Let $M$ be the midpoint of segment $\overline{AB}$ , and let $T$ lie on segment $\overline{AB}$ so that $AT \cdot AM = 100$ and $BT \cdot BM = 28$ . What is the length of segment $\overline{TM}$ ?

$\textbf{(A)}~4\qquad \textbf{(B)}~4.5\qquad \textbf{(C)}~5 \qquad \textbf{(D)}~5.5 \qquad \textbf{(E)}~6$

Problem 6

Equilateral triangle $ABC$ is partitioned into six smaller equilateral triangles and one smaller regular hexagon, as shown below. If the regular hexagon has area $12$ , what is the area of $\triangle ABC$ ?

$[asy] import graph; size(4.5cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ fill((-3.46, 6)--(-2.6, 6.5)--(-1.73, 6)--(-1.73, 5)--(-2.6, 4.5)--(-3.46, 5)--cycle, lightgrey); /* draw figures */ draw((-3.4641016151377544,6)--(-4.330127018922193,5.5)); draw((-4.330127018922193,5.5)--(-3.4641016151377544,5)); draw((-3.4641016151377544,5)--(-3.4641016151377544,6)); draw((-3.4641016151377544,5)--(-2.598076211353316,4.5)); draw((-2.598076211353316,4.5)--(-1.7320508075688772,5)); draw((-1.7320508075688772,5)--(-1.7320508075688772,6)); draw((-1.7320508075688772,6)--(-2.598076211353316,6.5)); draw((-2.598076211353316,6.5)--(-3.4641016151377544,6)); draw((-4.330127018922193,5.5)--(-4.330127018922193,3.5)); draw((-4.330127018922193,3.5)--(-2.598076211353316,4.5)); draw((-1.7320508075688772,5)--(-1.7320508075688772,2)); draw((-1.7320508075688772,2)--(-4.330127018922193,3.5)); draw((-1.7320508075688772,6)--(1.7320508075688772,4)); draw((1.7320508075688772,4)--(-1.7320508075688772,2)); draw((-4.330127018922193,5.5)--(-4.330127018922193,7.5)); draw((-4.330127018922193,7.5)--(-2.598076211353316,6.5)); draw((-4.330127018922193,3.5)--(-4.330127018922193,0.5)); draw((-4.330127018922193,0.5)--(-1.7320508075688772,2)); /* dot((-3.4641016151377544,6),dotstyle); dot((-4.330127018922193,5.5),dotstyle); dot((-3.4641016151377544,5),dotstyle); dot((-2.598076211353316,4.5),dotstyle); dot((-1.7320508075688772,5),dotstyle); dot((-1.7320508075688772,6),dotstyle); dot((-2.598076211353316,6.5),dotstyle); dot((-4.330127018922193,3.5),dotstyle); dot((-1.7320508075688772,2),dotstyle); */ dot((1.7320508075688772,4),dotstyle); label("$A$", (1.8087225843418686,4), E); dot((-4.330127018922193,7.5),dotstyle); label("$B$", (-4.266904757984109,7.678590535956637), NW); dot((-4.330127018922193,0.5),dotstyle); label("$C$", (-4.266904757984109,0.6655806893860435), SW * 2.5); label("$12$", (-2.6, 5.5)); [/asy]$

$\textbf{(A)}~ 72 \qquad \textbf{(B)}~ 84 \qquad \textbf{(C)}~ 98 \qquad \textbf{(D)}~ 128 \qquad \textbf{(E)}~ 147$

Problem 7

Let $N$ be the least positive integer that is divisible by at least $3$ odd primes and at least $4$ perfect squares. What is the sum of the squares of the digits of $N$ ?

$\textbf{(A)}~ 41 \qquad \textbf{(B)}~ 65 \qquad \textbf{(C)}~ 80 \qquad \textbf{(D)}~ 89 \qquad \textbf{(E)}~ 100$

Solution

Problem 8

Let $x$ be a real number with $\sin x \neq -1$ . What is the sum of the maximum and minimum possible values of $\frac{(\sin x + \cos x + 1)^{2}}{\sin x + 1}?$

$\textbf{(A)}~2 \qquad \textbf{(B)}~3 \qquad \textbf{(C)}~4 \qquad \textbf{(D)}~6 \qquad \textbf{(E)}~8$

Solution

Problem 9

Square $ABCD$ has side length $6$ and center $O$ . Points $E$ and $F$ lie in the plane, and $AOEF$ is a rectangle. Suppose that exactly $\tfrac{2}{3}$ of the area of $AOEF$ lies inside square $ABCD$ . What is the area of $\triangle CEF$ ?

$\textbf{(A)}~4\qquad\textbf{(B)}~3\sqrt{2}\qquad\textbf{(C)}~6\qquad\textbf{(D)}~4\sqrt{3}\qquad\textbf{(E)}~8$

Solution

Problem 10

Aubrey raced his younger brother Blair. Aubrey runs at a faster constant speed than Blair, so Blair started the race $40$ feet ahead of Aubrey. Aubrey caught up to Blair after $8$ seconds, finishing the race $90$ feet ahead of Blair and $5$ seconds earlier than Blair. How far did Aubrey run, in feet?

$\textbf{(A)}~454\qquad\textbf{(B)}~494\qquad\textbf{(C)}~518\qquad\textbf{(D)}~558\qquad\textbf{(E)}~598$

Solution

Problem 11

In regular tetrahedron $ABCD$ , points $E$ and $F$ lie on segments $\overline{AB}$ and $\overline{AC}$ , respectively, such that $BE = CF = 3$ . If $EF = 8$ , what is the area of $\triangle DEF$ ?

$\textbf{(A)}~32\qquad\textbf{(B)}~35\qquad\textbf{(C)}~36\qquad\textbf{(D)}~42\qquad\textbf{(E)}~48$

Solution

Problem 12

The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ 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 $(1, 2)$ ? $\begin{bmatrix}. & ? & . & . & . \\. & . & . & 48 & . \\ 12 & . & . & . & . \\ . & . & 16 & . & . \\ . & . & . & . & 0\end{bmatrix}$

$\textbf{(A)}~19\qquad\textbf{(B)}~24\qquad\textbf{(C)}~29\qquad\textbf{(D)}~34\qquad\textbf{(E)}~39$

Solution

Problem 13

Let $P(x)$ be a cubic polynomial with complex coefficients whose leading coefficient is real. Suppose $P(x)$ has two real roots and one complex root $z$ . If $|z - 1| = 10$ and $P(1) = 3 + 4i$ , where $i = \sqrt{-1}$ , what is the maximum possible value of $|z|$ ?

$\textbf{(A)}~\sqrt{89} \qquad \textbf{(B)}~10 \qquad \textbf{(C)}~\sqrt{113} \qquad \textbf{(D)}~\sqrt{117} \qquad \textbf{(E)}~11$

Solution

Problem 14

Points $X$ and $Y$ lie on sides $\overline{BC}$ and $\overline{CD}$ , respectively, of parallelogram $ABCD$ such that $\angle AXC = \angle AYC = 90^{\circ}$ . Suppose $BX = 5$ and $DY = 3$ , as shown. If $ABCD$ has perimeter $48$ , what is its area?

$[asy] import olympiad; import graph; size(8cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ pair A = (0, 0), B = (15, 0), C = (12, -6 * sqrt(2)), D = (-3, -6 * sqrt(2)); pair X = (15 - 3 * 5/9, -6 * sqrt(2) * 5 / 9); pair Y = (0, -6 * sqrt(2)); dot(A); dot(B); dot(C); dot(D); dot(X); dot(Y); draw(A--B--C--D--cycle); draw(A--X); draw(A--Y); draw(rightanglemark(A,X,C,15)); draw(rightanglemark(A,Y,C,15)); label("$A$", A, N * 1.5); label("$B$", B, N * 1.5); label("$C$", C, S * 1.5); label("$D$", D, S * 1.5); label("$X$", X, E * 1.5); label("$Y$", Y, S * 1.5); label("$3$", midpoint(D--Y), S * 1.5); label("$5$", midpoint(B--X), E * 1.5); [/asy]$

$\textbf{(A)}~40\sqrt{5}\qquad\textbf{(B)}~56\sqrt{3}\qquad\textbf{(C)}~48\sqrt{7}\qquad\textbf{(D)}~90\sqrt{2}\qquad\textbf{(E)}~60\sqrt{5}$

Solution

Problem 15

Suppose $p$ and $q$ are real numbers for which $\log_{p}(q) - \log_{2p}(q) =\frac{1}{3} \qquad \operatorname{and} \qquad \log_{2p}(q) - \log_{4p}(q) =\frac{1}{4}.$ What is the value of $\log_{4p}(q) - \log_{8p}(q)$ ?

$\textbf{(A)}~\frac{1}{6}\qquad\textbf{(B)}~\frac{7}{40}\qquad\textbf{(C)}~\frac{8}{45}\qquad\textbf{(D)}~\frac{7}{36}\qquad\textbf{(E)}~\frac{1}{5}$

Solution

Problem 16

How many subsets $S$ of $\{1, 2, 3, \cdots, 15\}$ with at least two elements satisfy the property that if $a$ and $b$ are distinct elements of $S$ , then $|a - b|$ is also an element of $S$ ?

$\textbf{(A)}~30 \qquad \textbf{(B)}~32 \qquad \textbf{(C)}~34 \qquad \textbf{(D)}~36 \qquad \textbf{(E)}~38$

Solution

Problem 17

Let $f(x)$ be a nonzero continuous function such that $f\left(\sqrt{x^{2} + y^{2}}\right) = f(x)f(y)$ for all real numbers $x$ and $y$ . If $f(2) \leq 2024$ , then how many integers in the set $\{-20, -19, \cdots, 20\}$ could be the value of $f(1)$ ?

$\textbf{(A)}~6\qquad\textbf{(B)}~7\qquad\textbf{(C)}~12\qquad\textbf{(D)}~13\qquad\textbf{(E)}~16$

Solution

Problem 18

Let $P_{1}$ and $P_{2}$ be distinct points in the plane, and for positive integers $n \geq 3$ , $P_{n}$ is constructed according to the following rules:

If $n$ is odd, then $P_{n}$ is obtained by rotating $P_{n - 2}$ about $P_{n - 1} ~ 60^{\circ}$ clockwise.
If $n$ is even, then $P_{n}$ is obtained by rotating $P_{n - 2}$ about $P_{n - 1} ~ 45^{\circ}$ clockwise.

What is the least positive integer $k > 1$ for which $P_{k} = P_{1}$ ?

$\textbf{(A)}~25 \qquad \textbf{(B)}~31\qquad \textbf{(C)}~37\qquad \textbf{(D)}~49 \qquad \textbf{(E)}~61$

Solution

Problem 19

Let $a$ , $b$ , and $c$ be pairwise relatively prime positive integers. Suppose one of these numbers is prime, and the other two are perfect squares. If $abc$ has $15a$ divisors and $a^{2}b^{2}c^{2}$ has $15b$ divisors, what is the least possible value of $a + b + c$ ?

$\textbf{(A)}~18\qquad\textbf{(B)}~44\qquad\textbf{(C)}~108\qquad\textbf{(D)}~141\qquad\textbf{(E)}~636$

Solution

Problem 20

The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1^{\prime\prime} \times 1^{\prime\prime}$ 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?

$[asy] size(6cm); for (int i=0; i<9; ++i) { draw((i,0)--(i,3),dotted); } for (int i=0; i<4; ++i){ draw((0,i)--(8,i),dotted); } for (int i=0; i<8; ++i) { for (int j=0; j<3; ++j) { if (j==1) { label("1",(i+0.5,1.5)); }}} [/asy]$

$\textbf{(A)}~130\qquad\textbf{(B)}~144\qquad\textbf{(C)}~146\qquad\textbf{(D)}~162\qquad\textbf{(E)}~196$

Solution

Problem 21

A graph is $\textit{symmetric}$ about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a, b, c, d)$ , where $|a|, |b|, |c|, |d| \leq 5$ and $c$ and $d$ are not both $0$ , is the graph of $y =\frac{ax + b}{cx + d}$ symmetric about the line $y = x$ ?

$\textbf{(A)}~1282\qquad\textbf{(B)}~1292\qquad\textbf{(C)}~1310\qquad\textbf{(D)}~1320\qquad\textbf{(E)}~1330$

Solution

Problem 22

Three circles of radius $6$ are mutually externally tangent, as shown below. For each pair of circles, construct the lines through their point of tangency that are tangent to the third circle. In total, this creates six new tangency points. If the area of the convex hexagon formed by these six points can be expressed as $a\sqrt{2} + b\sqrt{3}$ for integers $a$ and $b$ , what is $a + b$ ?

$[asy] import olympiad; size(150); defaultpen(linewidth(0.6) + fontsize(10)); pen dotstyle = black; real xmin = -8.903758953234952, xmax = 14.962621077942625, ymin = -5.903115312371685, ymax = 10.567409309904837; /* image dimensions */ /* draw figures */ draw(circle((0,0), 3), linewidth(1)); draw(circle((6,0), 3), linewidth(1)); draw(circle((3,5.196152422706632), 3), linewidth(1)); draw((0.5505102572168217,3.4641016151377544)--(3,0), linewidth(1)); draw((3,0)--(5.449489742783179,3.4641016151377544), linewidth(1)); /* dots */ dot((0.5505102572168217,3.4641016151377544),linewidth(4pt) + dotstyle); dot((5.449489742783179,3.4641016151377544),linewidth(4pt) + dotstyle); dot((5.724744871391589,2.987345747344081),linewidth(4pt) + dotstyle); dot((3.275255128608411,-1.2552949397752038),linewidth(4pt) + dotstyle); dot((0.27525512860841106,2.987345747344081),linewidth(4pt) + dotstyle); dot((2.724744871391589,-1.2552949397752038),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]$

$\textbf{(A)}~18\qquad\textbf{(B)}~24\qquad\textbf{(C)}~36\qquad\textbf{(D)}~42\qquad\textbf{(E)}~45$

Solution

Problem 23

In parallelogram $ABCD$ , let $\omega$ be the circle with diameter $\overline{AD}$ and suppose $P$ and $Q$ are points on $\omega$ such that both lines $BP$ and $BQ$ are tangent to $\omega$ . If $BC = 8$ , $BP = 3$ , and line $PQ$ bisects $\overline{CD}$ , what is $AC^{2}$ ?

$\textbf{(A)}~180\qquad\textbf{(B)}~181\qquad\textbf{(C)}~182\qquad\textbf{(D)}~183\qquad\textbf{(E)}~184$

Solution

Problem 24

There exist exactly four different complex numbers $z$ that satisfy the equation shown below: $\left|z + \overline{z}\left(\tfrac{3}{5} + \tfrac{4}{5}i\right)\right| = \left|z + \overline{z}\left(\tfrac{4}{5} + \tfrac{3}{5}i\right)\right| = 1.$ What is the area of the convex quadrilateral whose vertices are those four complex numbers $z$ in the complex plane?

$\textbf{(A)}~\frac{4\sqrt{21}}{3}\qquad\textbf{(B)}~\frac{9\sqrt{2}}{2}\qquad\textbf{(C)}~5\sqrt{2}\qquad\textbf{(D)}~2\sqrt{14}\qquad\textbf{(E)}~\frac{7\sqrt{5}}{2}$

Solution

Problem 25

The ellipse $2x^{2} + 3y^{2} = 7$ consists of all points $P$ in the coordinate plane satisfying $PF_{1} + PF_{2} = \lambda$ , for some points $F_{1}$ and $F_{2}$ and some constant $\lambda$ . Let $\mathbf{R}$ denote the set of all points $Q$ in the coordinate plane satisfying $\sqrt{QF_{1}^{2} + 1}\ + \sqrt{QF_{2}^{2} + 1} = \lambda$ What is the square of the area of the region bounded by $\mathbf{R}$ ?

$\textbf{(A)}~\frac{147\pi^{2}}{64}\qquad\textbf{(B)}~\frac{8\pi^{2}}{3}\qquad\textbf{(C)}~\frac{27\pi^{2}}{7}\qquad\textbf{(D)}~\frac{25\pi^{2}}{6}\qquad\textbf{(E)}~\frac{49\pi^{2}}{10}$

Solution

2024 AMC 12A (Problems • Answer Key • Resources)
Preceded by 2023 AMC 12B Problems	Followed by 2024 AMC 12B Problems
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All AMC 12 Problems and Solutions

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2024 AMC 12A Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

See also