Difference between revisions of "2024 AMC 12A Problems"

Revision as of 21:31, 20 March 2025

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.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25

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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Difference between revisions of "2024 AMC 12A Problems"

Revision as of 21:31, 20 March 2025

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

@@ Line 2: / Line 2: @@
 ==Problem 1==
+What is the value of <math>9901 \cdot 101 - 99 \cdot 10101</math>?
-What is the value of <math>9901\cdot101-99\cdot10101?</math>
 <math>\textbf{(A)}~2\qquad\textbf{(B)}~20\qquad\textbf{(C)}~200\qquad\textbf{(D)}~202\qquad\textbf{(E)}~2020</math>
@@ Line 10: / Line 9: @@
 ==Problem 2==
+Define <math>\blacktriangledown(a) = \sqrt{a - 1}</math> and <math>\blacktriangle(a) = \sqrt{a + 1}</math> for all real numbers <math>a</math>. What is the value of <cmath>\frac{\blacktriangledown(20 + \blacktriangle(2024))}{\blacktriangledown(\blacktriangle(24))}~?</cmath>
-A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form <math>T=aL+bG,</math> where <math>a</math> and <math>b</math> are constants, <math>T</math> is the time in minutes, <math>L</math> is the length of the trail in miles, and <math>G</math> is the altitude gain in feet. The model estimates that it will take <math>69</math> minutes to hike to the top if a trail is <math>1.5</math> miles long and ascends <math>800</math> feet, as well as if a trail is <math>1.2</math> miles long and ascends <math>1100</math> feet. How many minutes does the model estimates it will take to hike to the top if the trail is <math>4.2</math> miles long and ascends <math>4000</math> feet?
+<math>\textbf{(A)}~ 1 \qquad \textbf{(B)}~ 2 \qquad \textbf{(C)}~ 4 \qquad \textbf{(D)}~ 8 \qquad \textbf{(E)}~ 16</math>
-<math>\textbf{(A) }240\qquad\textbf{(B) }246\qquad\textbf{(C) }252\qquad\textbf{(D) }258\qquad\textbf{(E) }264</math>
 [[2024 AMC 12A Problems/Problem 2|Solution]]
 ==Problem 3==
+A square and an isosceles triangle are joined along an edge to form a pentagon <math>10</math> inches tall and <math>22</math> inches wide, as shown below. What is the perimeter of the pentagon, in inches?
-The number <math>2024</math> 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?
+<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>
-<math>\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24</math>
+<math>\textbf{(A)}~54\qquad \textbf{(B)}~56 \qquad \textbf{(C)}~62 \qquad \textbf{(D)}~64 \qquad \textbf{(E)}~66</math>
 [[2024 AMC 12A Problems/Problem 3|Solution]]
 ==Problem 4==
+A data set containing <math>20</math> numbers, some of which are <math>6</math>, has mean <math>45</math>. When all the <math>6</math>s are removed, the data set has mean <math>66</math>. How many <math>6</math>s were in the original data set?
-What is the least value of <math>n</math> such that <math>n!</math> is a multiple of <math>2024</math>?
+<math>\textbf{(A)}~4\qquad\textbf{(B)}~5\qquad\textbf{(C)}~6\qquad\textbf{(D)}~7\qquad\textbf{(E)}~8</math>
-<math>
-\textbf{(A) }11 \qquad
-\textbf{(B) }21 \qquad
-\textbf{(C) }22 \qquad
-\textbf{(D) }23 \qquad
-\textbf{(E) }253 \qquad
-</math>
 [[2024 AMC 12A Problems/Problem 4|Solution]]
 ==Problem 5==
+Let <math>M</math> be the midpoint of segment <math>\overline{AB}</math>, and let <math>T</math> lie on segment <math>\overline{AB}</math> so that <math>AT \cdot AM = 100</math> and <math>BT \cdot BM = 28</math>. What is the length of segment <math>\overline{TM}</math>?
-A data set containing <math>20</math> numbers, some of which are <math>6</math>, has mean <math>45</math>. When all the 6s are removed, the data set has mean <math>66</math>. How many 6s were in the original data set?
+<math>\textbf{(A)}~4\qquad \textbf{(B)}~4.5\qquad \textbf{(C)}~5 \qquad \textbf{(D)}~5.5 \qquad \textbf{(E)}~6</math>
-<math>\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8</math>
-[[2024 AMC 12A Problems/Problem 5|Solution]]
 ==Problem 6==
+Equilateral triangle <math>ABC</math> is partitioned into six smaller equilateral triangles and one smaller regular hexagon, as shown below.  If the regular hexagon has area <math>12</math>, what is the area of <math>\triangle ABC</math>?
-The product of three integers is <math>60</math>. What is the least possible positive sum of the three integers?
+<asy>
+import graph; size(4.5cm);
-<math>\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13</math>
+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>
-[[2024 AMC 12A Problems/Problem 6|Solution]]
+<math>\textbf{(A)}~ 72 \qquad \textbf{(B)}~ 84 \qquad \textbf{(C)}~ 98 \qquad \textbf{(D)}~ 128 \qquad \textbf{(E)}~ 147</math>
 ==Problem 7==
+Let <math>N</math> be the least positive integer that is divisible by at least <math>3</math> odd primes and at least <math>4</math> perfect squares. What is the sum of the squares of the digits of <math>N</math>?
-In <math>\Delta ABC</math>, <math>\angle ABC = 90^\circ</math> and <math>BA = BC = \sqrt{2}</math>. Points <math>P_1, P_2, \dots, P_{2024}</math> lie on hypotenuse <math>\overline{AC}</math> so that <math>AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C</math>. What is the length of the vector sum
+<math>\textbf{(A)}~ 41 \qquad \textbf{(B)}~ 65 \qquad \textbf{(C)}~ 80 \qquad \textbf{(D)}~ 89 \qquad \textbf{(E)}~ 100</math>
-<cmath> \overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \dots + \overrightarrow{BP_{2024}}? </cmath>
-<math>
-\textbf{(A) }1011 \qquad
-\textbf{(B) }1012 \qquad
-\textbf{(C) }2023 \qquad
-\textbf{(D) }2024 \qquad
-\textbf{(E) }2025 \qquad
-</math>
 [[2024 AMC 12A Problems/Problem 7|Solution]]
 ==Problem 8==
+Let <math>x</math> be a real number with <math>\sin x \neq -1</math>. What is the sum of the maximum and minimum possible values of <cmath>\frac{(\sin x + \cos x + 1)^{2}}{\sin x + 1}?</cmath>
-How many angles <math>\theta</math> with <math>0\le\theta\le2\pi</math> satisfy <math>\log(\sin(3\theta))+\log(\cos(2\theta))=0</math>?
+<math>\textbf{(A)}~2 \qquad \textbf{(B)}~3 \qquad \textbf{(C)}~4 \qquad \textbf{(D)}~6 \qquad \textbf{(E)}~8</math>
-<math> \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad </math>
 [[2024 AMC 12A Problems/Problem 8|Solution]]
 ==Problem 9==
+Square <math>ABCD</math> has side length <math>6</math> and center <math>O</math>. Points <math>E</math> and <math>F</math> lie in the plane, and <math>AOEF</math> is a rectangle. Suppose that exactly <math>\tfrac{2}{3}</math> of the area of <math>AOEF</math> lies inside square <math>ABCD</math>. What is the area of <math>\triangle CEF</math>?
-Let <math>M</math> be the greatest integer such that both <math>M + 1213</math> and <math>M + 3773</math> are perfect squares. What is the units digit of <math>M</math>?
+<math>\textbf{(A)}~4\qquad\textbf{(B)}~3\sqrt{2}\qquad\textbf{(C)}~6\qquad\textbf{(D)}~4\sqrt{3}\qquad\textbf{(E)}~8</math>
-<math>
-\textbf{(A) }1 \qquad
-\textbf{(B) }2 \qquad
-\textbf{(C) }3 \qquad
-\textbf{(D) }6 \qquad
-\textbf{(E) }8 \qquad
-</math>
 [[2024 AMC 12A Problems/Problem 9|Solution]]
 ==Problem 10==
+Aubrey raced his younger brother Blair. Aubrey runs at a faster constant speed than Blair, so Blair started the race <math>40</math> feet ahead of Aubrey. Aubrey caught up to Blair after <math>8</math> seconds, finishing the race <math>90</math> feet ahead of Blair and <math>5</math> seconds earlier than Blair. How far did Aubrey run, in feet?
-Let <math>\alpha</math> be the radian measure of the smallest angle in a <math>3{-}4{-}5</math> right triangle. Let <math>\beta</math> be the radian measure of the smallest angle in a <math>7{-}24{-}25</math> right triangle. In terms of <math>\alpha</math>, what is <math>\beta</math>?
+<math>\textbf{(A)}~454\qquad\textbf{(B)}~494\qquad\textbf{(C)}~518\qquad\textbf{(D)}~558\qquad\textbf{(E)}~598</math>
-<math>
-\textbf{(A) }\frac{\alpha}{3}\qquad
-\textbf{(B) }\alpha - \frac{\pi}{8}\qquad
-\textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad
-\textbf{(D) }\frac{\alpha}{2}\qquad
-\textbf{(E) }\pi - 4\alpha\qquad
-</math>
 [[2024 AMC 12A Problems/Problem 10|Solution]]
 ==Problem 11==
+In regular tetrahedron <math>ABCD</math>, points <math>E</math> and <math>F</math> lie on segments <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, such that <math>BE = CF = 3</math>. If <math>EF = 8</math>, what is the area of <math>\triangle DEF</math>?
-There are exactly <math>K</math> positive integers <math>b</math> with <math>5 \leq b \leq 2024</math> such that the base-<math>b</math> integer <math>2024_b</math> is divisible by <math>16</math> (where <math>16</math> is in base ten). What is the sum of the digits of <math>K</math>?
+<math>\textbf{(A)}~32\qquad\textbf{(B)}~35\qquad\textbf{(C)}~36\qquad\textbf{(D)}~42\qquad\textbf{(E)}~48</math>
-<math>\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21</math>
 [[2024 AMC 12A Problems/Problem 11|Solution]]
 ==Problem 12==
+The numbers, in order, of each row and the numbers, in order, of each column of a <math>5 \times 5</math> array of integers form an arithmetic progression of length <math>5</math>. The numbers in positions <math>(5, 5)</math>, <math>(2, 4)</math>, <math>(4, 3)</math>, and <math>(3, 1)</math> are <math>0</math>, <math>48</math>, <math>16</math>, and <math>12</math>, respectively. What number is in position <math>(1, 2)</math>? <cmath>\begin{bmatrix}. & ? & . & . & . \\. & . & . & 48 & . \\ 12 & . & . & . & . \\ . & . & 16 & . & . \\ . & . & . & . & 0\end{bmatrix}</cmath>
-The first three terms of a geometric sequence are the integers <math>a,\,720,</math> and <math>b,</math> where <math>a<720<b.</math> What is the sum of the digits of the least possible value of <math>b?</math>
+<math>\textbf{(A)}~19\qquad\textbf{(B)}~24\qquad\textbf{(C)}~29\qquad\textbf{(D)}~34\qquad\textbf{(E)}~39</math>
-<math>\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21</math>
 [[2024 AMC 12A Problems/Problem 12|Solution]]
 ==Problem 13==
+Let <math>P(x)</math> be a cubic polynomial with complex coefficients whose leading coefficient is real. Suppose <math>P(x)</math> has two real roots and one complex root <math>z</math>. If <math>|z - 1| = 10</math> and <math>P(1) = 3 + 4i</math>, where <math>i = \sqrt{-1}</math>, what is the maximum possible value of <math>|z|</math>?
-The graph of <math>y=e^{x+1}+e^{-x}-2</math> has an axis of symmetry. What is the reflection of the point <math>(-1,\tfrac{1}{2})</math> over this axis?
+<math>\textbf{(A)}~\sqrt{89} \qquad \textbf{(B)}~10 \qquad \textbf{(C)}~\sqrt{113} \qquad \textbf{(D)}~\sqrt{117} \qquad \textbf{(E)}~11</math>
-<math>\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\frac{1}{2}\right)</math>
 [[2024 AMC 12A Problems/Problem 13|Solution]]
 ==Problem 14==
+Points <math>X</math> and <math>Y</math> lie on sides <math>\overline{BC}</math> and <math>\overline{CD}</math>, respectively, of parallelogram <math>ABCD</math> such that <math>\angle AXC = \angle AYC = 90^{\circ}</math>. Suppose <math>BX = 5</math> and <math>DY = 3</math>, as shown. If <math>ABCD</math> has perimeter <math>48</math>, what is its area?
-The numbers, in order, of each row and the numbers, in order, of each column of a <math>5 \times 5</math> array of integers form an arithmetic progression of length <math>5{.}</math> The numbers in positions <math>(5, 5), \,(2,4),\,(4,3),</math> and <math>(3, 1)</math> are <math>0, 48, 16,</math> and <math>12{,}</math> respectively. What number is in position <math>(1, 2)?</math>
+<asy>
-<cmath> \begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}</cmath>
+import olympiad; import graph;
-<math>\textbf{(A) } 19 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 29 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 39</math>
+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>
+<math>\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}</math>
 [[2024 AMC 12A Problems/Problem 14|Solution]]
 ==Problem 15==
+Suppose <math>p</math> and <math>q</math> are real numbers for which <cmath>\log_{p}(q) - \log_{2p}(q) =\frac{1}{3} \qquad \operatorname{and} \qquad \log_{2p}(q) - \log_{4p}(q) =\frac{1}{4}.</cmath> What is the value of <math>\log_{4p}(q) - \log_{8p}(q)</math>?
-The roots of <math>x^3 + 2x^2 - x + 3</math> are <math>p, q,</math> and <math>r.</math> What is the value of <cmath>(p^2 + 4)(q^2 + 4)(r^2 + 4)?</cmath>
+<math>\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}</math>
-<math>\textbf{(A) } 64 \qquad \textbf{(B) } 75 \qquad \textbf{(C) } 100 \qquad \textbf{(D) } 125 \qquad \textbf{(E) } 144</math>
 [[2024 AMC 12A Problems/Problem 15|Solution]]
 ==Problem 16==
+How many subsets <math>S</math> of <math>\{1, 2, 3, \cdots, 15\}</math> with at least two elements satisfy the property that if <math>a</math> and <math>b</math> are distinct elements of <math>S</math>, then <math>|a - b|</math> is also an element of <math>S</math>?
-A set of <math>12</math> tokens ---- <math>3</math> red, <math>2</math> white, <math>1</math> blue, and <math>6</math> black ---- is to be distributed at random to <math>3</math> game players, <math>4</math> tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>?
+<math>\textbf{(A)}~30 \qquad \textbf{(B)}~32 \qquad \textbf{(C)}~34 \qquad \textbf{(D)}~36 \qquad \textbf{(E)}~38</math>
-<math>
-\textbf{(A) }387 \qquad
-\textbf{(B) }388 \qquad
-\textbf{(C) }389 \qquad
-\textbf{(D) }390 \qquad
-\textbf{(E) }391 \qquad
-</math>
 [[2024 AMC 12A Problems/Problem 16|Solution]]
 ==Problem 17==
+Let <math>f(x)</math> be a nonzero continuous function such that <cmath>f\left(\sqrt{x^{2} + y^{2}}\right) = f(x)f(y)</cmath> for all real numbers <math>x</math> and <math>y</math>. If <math>f(2) \leq 2024</math>, then how many integers in the set <math>\{-20, -19, \cdots, 20\}</math> could be the value of <math>f(1)</math>?
-Integers <math>a</math>, <math>b</math>, and <math>c</math> satisfy <math>ab + c = 100</math>, <math>bc + a = 87</math>, and <math>ca + b = 60</math>. What is <math>ab + bc + ca</math>?
+<math>\textbf{(A)}~6\qquad\textbf{(B)}~7\qquad\textbf{(C)}~12\qquad\textbf{(D)}~13\qquad\textbf{(E)}~16</math>
-<math>
-\textbf{(A) }212 \qquad
-\textbf{(B) }247 \qquad
-\textbf{(C) }258 \qquad
-\textbf{(D) }276 \qquad
-\textbf{(E) }284 \qquad
-</math>
 [[2024 AMC 12A Problems/Problem 17|Solution]]
 ==Problem 18==
+Let <math>P_{1}</math> and <math>P_{2}</math> be distinct points in the plane, and for positive integers <math>n \geq 3</math>, <math>P_{n}</math> is constructed according to the following rules:
-On top of a rectangular card with sides of length <math>1</math> and <math>2+\sqrt{3}</math>, an identical card is placed so that two of their diagonals line up, as shown (<math>\overline{AC}</math>, in this case).
+* If <math>n</math> is odd, then <math>P_{n}</math> is obtained by rotating <math>P_{n - 2}</math> about <math>P_{n - 1} ~ 60^{\circ}</math> clockwise.
+* If <math>n</math> is even, then <math>P_{n}</math> is obtained by rotating <math>P_{n - 2}</math> about <math>P_{n - 1} ~ 45^{\circ}</math> clockwise.
-<asy>
+What is the least positive integer <math>k > 1</math> for which <math>P_{k} = P_{1}</math>?
-defaultpen(fontsize(12)+0.85); size(150);
-real h=2.25;
-pair C=origin,B=(0,h),A=(1,h),D=(1,0),Dp=reflect(A,C)*D,Bp=reflect(A,C)*B;
-pair L=extension(A,Dp,B,C),R=extension(Bp,C,A,D);
-draw(L--B--A--Dp--C--Bp--A);
-draw(C--D--R);
-draw(L--C^^R--A,dashed+0.6);
-draw(A--C,black+0.6);
-dot("$C$",C,2*dir(C-R)); dot("$A$",A,1.5*dir(A-L)); dot("$B$",B,dir(B-R));
-</asy>
-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 <math>B</math> in the figure?
-<math>\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }\text{No new vertex will land on }B.</math>
+<math>\textbf{(A)}~25 \qquad \textbf{(B)}~31\qquad \textbf{(C)}~37\qquad \textbf{(D)}~49 \qquad \textbf{(E)}~61</math>
 [[2024 AMC 12A Problems/Problem 18|Solution]]
 ==Problem 19==
+Let <math>a</math>, <math>b</math>, and <math>c</math> be pairwise relatively prime positive integers. Suppose one of these numbers is prime, and the other two are perfect squares. If <math>abc</math> has <math>15a</math> divisors and <math>a^{2}b^{2}c^{2}</math> has <math>15b</math> divisors, what is the least possible value of <math>a + b + c</math>?
-Cyclic quadrilateral <math>ABCD</math> has lengths <math>BC=CD=3</math> and <math>DA=5</math> with <math>\angle CDA=120^\circ</math>. What is the length of the shorter diagonal of <math>ABCD</math>?
+<math>\textbf{(A)}~18\qquad\textbf{(B)}~44\qquad\textbf{(C)}~108\qquad\textbf{(D)}~141\qquad\textbf{(E)}~636</math>
-<math>
-\textbf{(A) }\frac{31}7 \qquad
-\textbf{(B) }\frac{33}7 \qquad
-\textbf{(C) }5 \qquad
-\textbf{(D) }\frac{39}7 \qquad
-\textbf{(E) }\frac{41}7 \qquad
-</math>
 [[2024 AMC 12A Problems/Problem 19|Solution]]
 ==Problem 20==
+The figure below shows a dotted grid <math>8</math> cells wide and <math>3</math> cells tall consisting of <math>1^{\prime\prime} \times 1^{\prime\prime}</math> squares. Carl places <math>1</math>-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?
-Points <math>P</math> and <math>Q</math> are chosen uniformly and independently at random on sides <math>\overline {AB}</math> and <math>\overline{AC},</math> respectively, of equilateral triangle <math>\Delta ABC.</math> Which of the following intervals contains the probability that the area of <math>\triangle APQ</math> is less than half the area of <math>\triangle ABC?</math>
-<math>\textbf{(A) } \left[\frac 38, \frac 12\right] \qquad \textbf{(B) } \left(\frac 12, \frac 23\right] \qquad \textbf{(C) } \left(\frac 23, \frac 34\right] \qquad \textbf{(D) } \left(\frac 34, \frac 78\right] \qquad \textbf{(E) } \left(\frac 78, 1\right]</math>
-[[2024 AMC 12A Problems/Problem 20|Solution]]
-==Problem 21==
-Suppose that <math>a_1 = 2</math> and the sequence <math>(a_n)</math> satisfies the recurrence relation <cmath>\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}</cmath>for all <math>n \ge 2.</math> What is the greatest integer less than or equal to <cmath>\sum^{100}_{n=1} a_n^2?</cmath>
-<math>\textbf{(A) } 338{,}550 \qquad \textbf{(B) } 338{,}551 \qquad \textbf{(C) } 338{,}552 \qquad \textbf{(D) } 338{,}553 \qquad \textbf{(E) } 338{,}554</math>
-[[2024 AMC 12A Problems/Problem 21|Solution]]
-==Problem 22==
-The figure below shows a dotted grid <math>8</math> cells wide and <math>3</math> cells tall consisting of <math>1''\times1''</math> squares. Carl places <math>1</math>-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>
@@ Line 242: / Line 243: @@
 </asy>
-<math>\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196</math>
+<math>\textbf{(A)}~130\qquad\textbf{(B)}~144\qquad\textbf{(C)}~146\qquad\textbf{(D)}~162\qquad\textbf{(E)}~196</math>
+[[2024 AMC 12A Problems/Problem 20|Solution]]
+==Problem 21==
+A graph is <math>\textit{symmetric}</math> about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers <math>(a, b, c, d)</math>, where <math>|a|, |b|, |c|, |d| \leq 5</math> and <math>c</math> and <math>d</math> are not both <math>0</math>, is the graph of <cmath>y =\frac{ax + b}{cx + d}</cmath> symmetric about the line <math>y = x</math>?
+<math>\textbf{(A)}~1282\qquad\textbf{(B)}~1292\qquad\textbf{(C)}~1310\qquad\textbf{(D)}~1320\qquad\textbf{(E)}~1330</math>
+[[2024 AMC 12A Problems/Problem 21|Solution]]
+==Problem 22==
+Three circles of radius <math>6</math> 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 <math>a\sqrt{2} + b\sqrt{3}</math> for integers <math>a</math> and <math>b</math>, what is <math>a + b</math>?
+<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>
+<math>\textbf{(A)}~18\qquad\textbf{(B)}~24\qquad\textbf{(C)}~36\qquad\textbf{(D)}~42\qquad\textbf{(E)}~45</math>
 [[2024 AMC 12A Problems/Problem 22|Solution]]
 ==Problem 23==
+In parallelogram <math>ABCD</math>, let <math>\omega</math> be the circle with diameter <math>\overline{AD}</math> and suppose <math>P</math> and <math>Q</math> are points on <math>\omega</math> such that both lines <math>BP</math> and <math>BQ</math> are tangent to <math>\omega</math>. If <math>BC = 8</math>, <math>BP = 3</math>, and line <math>PQ</math> bisects <math>\overline{CD}</math>, what is <math>AC^{2}</math>?
-What is the value of
+<math>\textbf{(A)}~180\qquad\textbf{(B)}~181\qquad\textbf{(C)}~182\qquad\textbf{(D)}~183\qquad\textbf{(E)}~184</math>
-<cmath>\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16}~ + ~ \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16} ~+~\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16} ~+~ \tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?</cmath>
-<math>\textbf{(A) } 28 \qquad \textbf{(B) } 68 \qquad \textbf{(C) } 70 \qquad \textbf{(D) } 72 \qquad \textbf{(E) } 84</math>
 [[2024 AMC 12A Problems/Problem 23|Solution]]
 ==Problem 24==
+There exist exactly four different complex numbers <math>z</math> that satisfy the equation shown below: <cmath>\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.</cmath> What is the area of the convex quadrilateral whose vertices are those four complex numbers <math>z</math> in the complex plane?
-A <math>\textit{disphenoid}</math> 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?
+<math>\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}</math>
-<math>\textbf{(A) }\sqrt{3}\qquad\textbf{(B) }3\sqrt{15}\qquad\textbf{(C) }15\qquad\textbf{(D) }15\sqrt{7}\qquad\textbf{(E) }24\sqrt{6}</math>
 [[2024 AMC 12A Problems/Problem 24|Solution]]
 ==Problem 25==
+The ellipse <math>2x^{2} + 3y^{2} = 7</math> consists of all points <math>P</math> in the coordinate plane satisfying <math>PF_{1} + PF_{2} = \lambda</math>, for some points <math>F_{1}</math> and <math>F_{2}</math> and some constant <math>\lambda</math>. Let <math>\mathbf{R}</math> denote the set of all points <math>Q</math> in the coordinate plane satisfying <cmath>\sqrt{QF_{1}^{2} + 1}\ + \sqrt{QF_{2}^{2} + 1} = \lambda</cmath> What is the square of the area of the region bounded by <math>\mathbf{R}</math>?
-A graph is <math>\textit{symmetric}</math> about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers <math>(a,b,c,d)</math>, where <math>|a|,|b|,|c|,|d|\le5</math> and <math>c</math> and <math>d</math> are not both <math>0</math>, is the graph of <cmath>y=\frac{ax+b}{cx+d}</cmath>symmetric about the line <math>y=x</math>?
+<math>\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}</math>
-<math>\textbf{(A) }1282\qquad\textbf{(B) }1292\qquad\textbf{(C) }1310\qquad\textbf{(D) }1320\qquad\textbf{(E) }1330</math>
 [[2024 AMC 12A Problems/Problem 25|Solution]]