Difference between revisions of "2024 AMC 10A Problems"

Revision as of 21:31, 20 March 2025

2024 AMC 10A (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 SAT 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

What is the sum of the digits of the smallest prime that can be written as a sum of $5$ distinct primes?

$\textbf{(A)}~5\qquad\textbf{(B)}~7\qquad\textbf{(C)}~9\qquad\textbf{(D)}~10\qquad\textbf{(E)}~11$

Solution

Problem 4

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 5

Andrea is taking a series of several exams. If Andrea earns $61$ points on her next exam, her average score will decrease by $3$ points. If she instead earns $93$ points on her next exam, her average score will increase by $1$ point. How many points should Andrea earn on her next exam to keep her average score constant?

$\textbf{(A)}~80 \qquad\textbf{(B)}~82 \qquad\textbf{(C)}~83 \qquad\textbf{(D)}~85 \qquad\textbf{(E)}~86$

Solution

Problem 6

How many ordered pairs $(m, n)$ of positive integers exist such that $m$ is a factor of $54$ and $mn$ is a factor of $70$ ?

$\textbf{(A)}~2 \qquad\textbf{(B)}~4 \qquad\textbf{(C)}~10 \qquad\textbf{(D)}~12 \qquad\textbf{(E)}~16$

Solution

Problem 7

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 $\overline{TM}$ ?

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

Solution

Problem 8

In how many ways can $6$ juniors and $6$ seniors form $3$ disjoint teams of $4$ people so that each team has $2$ juniors and $2$ seniors?

$\textbf{(A)}~720\qquad\textbf{(B)}~1350\qquad\textbf{(C)}~2700\qquad\textbf{(D)}~3280\qquad\textbf{(E)}~8100$

Solution

Problem 9

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 10

Let $\mathcal{K}$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt{3}$ along a common hypotenuse. Eight copies of $\mathcal{K}$ are used to form the polygon shown below. What is the area of triangle $\triangle ABC$ ?

$\textbf{(A)}~2 + 3\sqrt{3} \qquad\textbf{(B)}~\frac{9\sqrt{3}}{2} \qquad\textbf{(C)}~\frac{10 + 8\sqrt{3}}{3} \qquad\textbf{(D)}~8 \qquad\textbf{(E)}~5\sqrt{3}$

Solution

Problem 11

If $x$ and $y$ are real numbers satisfying $x + \tfrac{x}{y} = 2$ and $y + \tfrac{y}{x} = 6$ , what is the value of $x + y$ ?

$\textbf{(A)}~\frac{101}{28} \qquad\textbf{(B)}~\frac{42}{11} \qquad\textbf{(C)}~\frac{30}{7} \qquad\textbf{(D)}~\frac{14}{3} \qquad\textbf{(E)}~\frac{110}{21}$

Solution

Problem 12

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 13

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 14

All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is the length $AB$ ?

$\textbf{(A)}~4 + 4\sqrt{5}\qquad\textbf{(B)}~10\sqrt{2}\qquad\textbf{(C)}~5 + 5\sqrt{5}\qquad\textbf{(D)}~10\sqrt[4]{8}\qquad\textbf{(E)}~20$

Solution

Problem 15

Let $S$ be a subset of $\{1, 2, 3, \cdots, 2024\}$ such that the following two conditions hold:

If $x$ and $y$ are distinct elements of $S$ , then $|x - y| > 2$ .
If $x$ and $y$ are distinct odd elements of $S$ , then $|x - y| > 6$ .

What is the maximum possible number of elements in $S$ ?

$\textbf{(A)}~436\qquad\textbf{(B)}~506\qquad\textbf{(C)}~608\qquad\textbf{(D)}~654\qquad\textbf{(E)}~675$

Solution

Problem 16

In how many ways can the integers $1$ , $2$ , $3$ , $4$ , $5$ , and $6$ be arranged in a line so that the following statement is true? If $2$ is not adjacent to $3$ , then $3$ is not adjacent to $4$ .

$\textbf{(A)}~480 \qquad\textbf{(B)}~504 \qquad\textbf{(C)}~528 \qquad\textbf{(D)}~572 \qquad\textbf{(E)}~600$

Solution

Problem 17

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 18

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 19

A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^{+}, A^{-}, B^{+}, B^{-}, C^{+}$ , and $C^{-}$ is rolled. Suppose the bee occupies the point $(a, b, c)$ . If the die shows $A^{+}$ , then the bee moves to the point $(a + 1, b, c)$ and if the die shows $A^{-}$ , then the bee moves to the point $(a - 1, b, c)$ . Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0, 0, 0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?

$\textbf{(A)}~\frac{1}{54}\qquad\textbf{(B)}~\frac{7}{54}\qquad\textbf{(C)}~\frac{1}{6}\qquad\textbf{(D)}~\frac{5}{18}\qquad\textbf{(E)}~\frac{2}{5}$

Solution

Problem 20

Point $X$ lies outside regular pentagon $ABCDE$ so that $\triangle BXE$ is an equilateral triangle, as shown below. What is the degree measure of acute angle $\angle CXD$ ?

$[asy] import graph; size(7cm); real labelscalefactor = 0.75; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -1.089556028373145, xmax = 2.738320502950249, ymin = -1.3143096399343266, ymax = 1.2691431521185288; /* image dimensions */ pen qqwuqq = rgb(0,0.39215686274509803,0); filldraw(arc((1.9562952014676112,0),0.25,168,192)--(1.9562952014676112,0)--cycle, mediumgrey); /* draw figures */ draw((-0.8090169943749473,0.5877852522924731)--(0.30901699437494745,0.9510565162951535)); draw((0.30901699437494745,0.9510565162951535)--(1,0)); draw((1,0)--(0.3090169943749473,-0.9510565162951536)); draw((0.3090169943749473,-0.9510565162951536)--(-0.8090169943749475,-0.587785252292473)); draw((-0.8090169943749475,-0.587785252292473)--(-0.8090169943749473,0.5877852522924731)); draw((0.30901699437494745,0.9510565162951535)--(0.3090169943749473,-0.9510565162951536)); draw((0.3090169943749473,-0.9510565162951536)--(1.9562952014676112,0)); draw((1.9562952014676112,0)--(0.30901699437494745,0.9510565162951535)); draw((-0.8090169943749473,0.5877852522924731)--(1.9562952014676112,0), dashed); draw((1.9562952014676112,0)--(-0.8090169943749475,-0.587785252292473), dashed); /* dots and labels */ dot((1,0),linewidth(4pt) + dotstyle); label("$A$", (1.013592864312142,0), E * labelscalefactor); dot((0.30901699437494745,0.9510565162951535),linewidth(4pt) + dotstyle); label("$B$", (0.26,1), NE * labelscalefactor); dot((-0.8090169943749473,0.5877852522924731),linewidth(4pt) + dotstyle); label("$C$", (-0.82,0.64), NW * 0.25); dot((-0.8090169943749475,-0.587785252292473),linewidth(4pt) + dotstyle); label("$D$", (-0.82,-0.64), SW * 0.25); dot((0.3090169943749473,-0.9510565162951536),linewidth(4pt) + dotstyle); label("$E$", (0.26,-1), SE * labelscalefactor); dot((1.9562952014676112,0),linewidth(4pt) + dotstyle); label("$X$", (1.9705619971429902, 0), E * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$

$\textbf{(A)}~18^{\circ}\qquad\textbf{(B)}~19.5^{\circ}\qquad\textbf{(C)}~21^{\circ}\qquad\textbf{(D)}~22.5^{\circ}\qquad\textbf{(E)}~24^{\circ}$

Solution

Problem 21

A fair six-sided die is repeatedly rolled until the same number is rolled twice in a row. What is the probability that the last number rolled is equal to the first number rolled?

$\textbf{(A)}~\frac{17}{72} \qquad\textbf{(B)}~\frac{4}{15} \qquad\textbf{(C)}~\frac{5}{18} \qquad\textbf{(D)}~\frac{2}{7} \qquad\textbf{(E)}~\frac{3}{10}$

Solution

Problem 22

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 23

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 24

There exists a unique two-digit prime number $p$ such that both $4^{28} - 15$ and $3^{28} - 20$ are divisible by $p$ . What is the sum of the digits of $p$ ?

$\textbf{(A)}~10\qquad\textbf{(B)}~11\qquad\textbf{(C)}~13\qquad\textbf{(D)}~14\qquad\textbf{(E)}~16$

Solution

Problem 25

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

2024 AMC 10A (Problems • Answer Key • Resources)
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Difference between revisions of "2024 AMC 10A 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 10A Problems/Problem 2|Solution]]
 ==Problem 3==
 What is the sum of the digits of the smallest prime that can be written as a sum of <math>5</math> distinct primes?
-<math>\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }13</math>
+<math>\textbf{(A)}~5\qquad\textbf{(B)}~7\qquad\textbf{(C)}~9\qquad\textbf{(D)}~10\qquad\textbf{(E)}~11</math>
 [[2024 AMC 10A Problems/Problem 3|Solution]]
 ==Problem 4==
+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 10A Problems/Problem 4|Solution]]
 ==Problem 5==
+Andrea is taking a series of several exams.  If Andrea earns <math>61</math> points on her next exam, her average score will decrease by <math>3</math> points.  If she instead earns <math>93</math> points on her next exam, her average score will increase by <math>1</math> point.  How many points should Andrea earn on her next exam to keep her average score constant?
-What is the least value of <math>n</math> such that <math>n!</math> is a multiple of <math>2024</math>?
+<math>\textbf{(A)}~80 \qquad\textbf{(B)}~82 \qquad\textbf{(C)}~83 \qquad\textbf{(D)}~85 \qquad\textbf{(E)}~86</math>
-<math>\textbf{(A) } 11\qquad\textbf{(B) } 21\qquad\textbf{(C) } 22\qquad\textbf{(D) } 23\qquad\textbf{(E) } 253</math>
 [[2024 AMC 10A Problems/Problem 5|Solution]]
 ==Problem 6==
+How many ordered pairs <math>(m, n)</math> of positive integers exist such that <math>m</math> is a factor of <math>54</math> and <math>mn</math> is a factor of <math>70</math>?
-What is the minimum number of successive swaps of adjacent letters in the string <math>ABCDEF</math> that are needed to change the string to <math>FEDCBA?</math> (For example, <math>3</math> swaps are required to change <math>ABC</math> to <math>CBA;</math> one such sequence of swaps is
+<math>\textbf{(A)}~2 \qquad\textbf{(B)}~4 \qquad\textbf{(C)}~10 \qquad\textbf{(D)}~12 \qquad\textbf{(E)}~16</math>
-<math>ABC\to BAC\to BCA\to CBA.</math>)
-<math>\textbf{(A)}~6\qquad\textbf{(B)}~10\qquad\textbf{(C)}~12\qquad\textbf{(D)}~15\qquad\textbf{(E)}~24</math>
 [[2024 AMC 10A Problems/Problem 6|Solution]]
 ==Problem 7==
+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>?
-The product of three integers is <math>60</math>. What is the least possible positive sum of the
+<math>\textbf{(A)}~4\qquad \textbf{(B)}~4.5\qquad \textbf{(C)}~5 \qquad \textbf{(D)}~5.5 \qquad \textbf{(E)}~6</math>
-three integers?
-<math>\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }13</math>
 [[2024 AMC 10A Problems/Problem 7|Solution]]
 ==Problem 8==
+In how many ways can <math>6</math> juniors and <math>6</math> seniors form <math>3</math> disjoint teams of <math>4</math> people so
+that each team has <math>2</math> juniors and <math>2</math> seniors?
-Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at <math>1:00 PM</math> and were able to pack <math>4</math>, <math>3</math>, and <math>3</math> packages, respectively, every <math>3</math> minutes. At some later time, Daria joined the group, and Daria was able to pack <math>5</math> packages every <math>4</math> minutes. Together, they finished packing <math>450</math> packages at exactly <math>2:45 PM</math>. At what time did Daria join the group?
+<math>\textbf{(A)}~720\qquad\textbf{(B)}~1350\qquad\textbf{(C)}~2700\qquad\textbf{(D)}~3280\qquad\textbf{(E)}~8100</math>
-<math>\textbf{(A) }1:25\text{ PM}\qquad\textbf{(B) }1:35\text{ PM}\qquad\textbf{(C) }1:45\text{ PM}\qquad\textbf{(D) }1:55\text{ PM}\qquad\textbf{(E) }2:05\text{ PM}</math>
 [[2024 AMC 10A Problems/Problem 8|Solution]]
 ==Problem 9==
+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 how many ways can <math>6</math> juniors and <math>6</math> seniors form <math>3</math> disjoint teams of <math>4</math> people so
+<math>\textbf{(A)}~ 41 \qquad \textbf{(B)}~ 65 \qquad \textbf{(C)}~ 80 \qquad \textbf{(D)}~ 89 \qquad \textbf{(E)}~ 100</math>
-that each team has <math>2</math> juniors and <math>2</math> seniors?
-<math>\textbf{(A) }720\qquad\textbf{(B) }1350\qquad\textbf{(C) }2700\qquad\textbf{(D) }3280\qquad\textbf{(E) }8100</math>
 [[2024 AMC 10A Problems/Problem 9|Solution]]
 ==Problem 10==
+Let <math>\mathcal{K}</math> be the kite formed by joining two right triangles with legs <math>1</math> and <math>\sqrt{3}</math> along a common hypotenuse. Eight copies of <math>\mathcal{K}</math> are used to form the polygon shown below. What is the area of triangle <math>\triangle ABC</math>?
+[[File:Screenshot_2024-11-08_3.23.29_PM.png|400px|center]]
-Consider the following operation. Given a positive integer <math>n</math>, if <math>n</math> is a multiple of <math>3</math>, then you replace <math>n</math> by <math>\frac{n}{3}</math>. If <math>n</math> is not a multiple of <math>3</math>, then you replace <math>n</math> by <math>n+10</math>. For example, beginning with <math>n=4</math>, this procedure gives <math>4\to14\to24\to8\to18\to6\to2\to12\to\cdots</math>. Suppose you start with <math>n=100</math>. What value results if you perform this operation exactly <math>100</math> times?
+<math>\textbf{(A)}~2 + 3\sqrt{3} \qquad\textbf{(B)}~\frac{9\sqrt{3}}{2} \qquad\textbf{(C)}~\frac{10 + 8\sqrt{3}}{3} \qquad\textbf{(D)}~8 \qquad\textbf{(E)}~5\sqrt{3}</math>
-<math>\textbf{(A) }10\qquad\textbf{(B) }20\qquad\textbf{(C) }30\qquad\textbf{(D) }40\qquad\textbf{(E) }50</math>
 [[2024 AMC 10A Problems/Problem 10|Solution]]
 ==Problem 11==
+If <math>x</math> and <math>y</math> are real numbers satisfying <math>x + \tfrac{x}{y} = 2</math> and <math>y + \tfrac{y}{x} = 6</math>, what is the value of <math>x + y</math>?
-How many ordered pairs of integers <math>(m, n)</math> satisfy <math>\sqrt{n^2 - 49} = m</math>?
+<math>\textbf{(A)}~\frac{101}{28} \qquad\textbf{(B)}~\frac{42}{11} \qquad\textbf{(C)}~\frac{30}{7} \qquad\textbf{(D)}~\frac{14}{3} \qquad\textbf{(E)}~\frac{110}{21}</math>
-<math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~\text{infinitely many}</math>
 [[2024 AMC 10A Problems/Problem 11|Solution]]
 ==Problem 12==
+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>?
-Zelda played the ''Adventures of Math'' game on August 1 and scored <math>1,700</math> points. She continued to play daily over the next <math>5</math> days. The bar chart below shows the daily change in her score compared to the day before. (For example, Zelda's score on August 2 was <math>1,700 + 80 = 1,780</math> points.) What was Zelda's average score in points over the <math>6</math> days?[[File:Screenshot_2024-11-08_1.51.51_PM.png]]
+<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)}~1700\qquad\textbf{(B)}~1702\qquad\textbf{(C)}~1703\qquad\textbf{(D)}~1713\qquad\textbf{(E)}~1715</math>
 [[2024 AMC 10A Problems/Problem 12|Solution]]
 ==Problem 13==
-Two transformations are said to commute if applying the first followed by the second
+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?
-gives the same result as applying the second followed by the first. Consider these
-four transformations of the coordinate plane:
-* a translation <math>2</math> units to the right,
-* a <math>90^{\circ}</math>-rotation counterclockwise about the origin,
-* a reflection across the <math>x</math>-axis, and
-* a dilation centered at the origin with scale factor <math>2.</math>
-Of the <math>6</math> pairs of distinct transformations from this list, how many commute?
-<math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5</math>
+<math>\textbf{(A)}~454\qquad\textbf{(B)}~494\qquad\textbf{(C)}~518\qquad\textbf{(D)}~558\qquad\textbf{(E)}~598</math>
 [[2024 AMC 10A Problems/Problem 13|Solution]]
 ==Problem 14==
+All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is the length <math>AB</math>?
-One side of an equilateral triangle of height <math>24</math> lies on line <math>\ell</math>. A circle of radius <math>12</math> is tangent to line <math>\ell</math> and is externally tangent to the triangle. The area of the region exterior to the triangle and the circle and bounded by the triangle, the circle, and line <math>\ell</math> can be written as <math>a \sqrt{b} - c \pi</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers and <math>b</math> is not divisible by the square of any prime. What is <math>a + b + c</math>?
+[[File:Screenshot 2024-11-08 2.08.49 PM.png|350px|center]]
-<math>\textbf{(A)}~72\qquad\textbf{(B)}~73\qquad\textbf{(C)}~74\qquad\textbf{(D)}~75\qquad\textbf{(E)}~76</math>
+<math>\textbf{(A)}~4 + 4\sqrt{5}\qquad\textbf{(B)}~10\sqrt{2}\qquad\textbf{(C)}~5 + 5\sqrt{5}\qquad\textbf{(D)}~10\sqrt[4]{8}\qquad\textbf{(E)}~20</math>
 [[2024 AMC 10A Problems/Problem 14|Solution]]
 ==Problem 15==
+Let <math>S</math> be a subset of <math>\{1, 2, 3, \cdots, 2024\}</math> such that the following two conditions hold:
-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>?
+* If <math>x</math> and <math>y</math> are distinct elements of <math>S</math>, then <math>|x - y| > 2</math>.
+* If <math>x</math> and <math>y</math> are distinct odd elements of <math>S</math>, then <math>|x - y| > 6</math>.
-<math>\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8</math>
+What is the maximum possible number of elements in <math>S</math>?
+<math>\textbf{(A)}~436\qquad\textbf{(B)}~506\qquad\textbf{(C)}~608\qquad\textbf{(D)}~654\qquad\textbf{(E)}~675</math>
 [[2024 AMC 10A Problems/Problem 15|Solution]]
 ==Problem 16==
+In how many ways can the integers <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, and <math>6</math> be arranged in a line so that the following statement is true? If <math>2</math> is not adjacent to <math>3</math>, then <math>3</math> is not adjacent to <math>4</math>.
-All of the rectangles in the figure below, which is drawn to scale, are similar to the enclosing rectangle. Each number represents the area of the rectangle. What is length <math>AB</math>? <math>\newline</math>
+<math>\textbf{(A)}~480 \qquad\textbf{(B)}~504 \qquad\textbf{(C)}~528 \qquad\textbf{(D)}~572 \qquad\textbf{(E)}~600</math>
-[[File:Screenshot 2024-11-08 2.08.49 PM.png]]
-<math>\textbf{(A) }4+4\sqrt5\qquad\textbf{(B) }10\sqrt2\qquad\textbf{(C) }5+5\sqrt5\qquad\textbf{(D) }10\sqrt[4]{8}\qquad\textbf{(E) }20</math>
 [[2024 AMC 10A Problems/Problem 16|Solution]]
 ==Problem 17==
+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> $[\begin{matrix} . & ? & . & . & . \\ . & . & . & 48 & . \\ 12 & . & . & . & . \\ . & . & 16 & . & . \\ . & . & . & . & 0 \end{matrix}]$ </cmath>
-Two teams are in a best-two-out-of-three playoff: the teams will play at most <math>3</math> games, and the winner of the playoff is the first team to win <math>2</math> games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a <math>\frac{2}{3}</math> chance of winning at home, and its probability of winning when playing away from home is <math>p</math>. Outcomes of the games are independent. The probability that Team A wins the playoff is <math>\frac{1}{2}</math>. Then <math>p</math> can be written in the form <math>\frac{1}{2}(m - \sqrt{n})</math>, where <math>m</math> and <math>n</math> are positive integers. What is <math>m+n</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)}~10\qquad\textbf{(B)}~11\qquad\textbf{(C)}~12\qquad\textbf{(D)}~13\qquad\textbf{(E)}~14</math>
 [[2024 AMC 10A Problems/Problem 17|Solution]]
 ==Problem 18==
+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?
-There are exactly <math>K</math> positive integers <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>?
+<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>
-<math>\textbf{(A)}~16\qquad\textbf{(B)}~17\qquad\textbf{(C)}~18\qquad\textbf{(D)}~20\qquad\textbf{(E)}~21</math>
+<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 10A Problems/Problem 18|Solution]]
 ==Problem 19==
+A bee is moving in three-dimensional space. A fair six-sided die with faces labeled <math>A^{+}, A^{-}, B^{+}, B^{-}, C^{+}</math>, and <math>C^{-}</math> is rolled. Suppose the bee occupies the point <math>(a, b, c)</math>. If the die shows <math>A^{+}</math>, then the bee moves to the point <math>(a + 1, b, c)</math> and if the die shows <math>A^{-}</math>, then the bee moves to the point <math>(a - 1, b, c)</math>. Analogous moves are made with the other four outcomes. Suppose the bee starts at the point <math>(0, 0, 0)</math> and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
-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)}~\frac{1}{54}\qquad\textbf{(B)}~\frac{7}{54}\qquad\textbf{(C)}~\frac{1}{6}\qquad\textbf{(D)}~\frac{5}{18}\qquad\textbf{(E)}~\frac{2}{5}</math>
-<math>\textbf{(A) } 9\qquad\textbf{(B) } 12\qquad\textbf{(C) } 16\qquad\textbf{(D) } 18\qquad\textbf{(E) } 21</math>
 [[2024 AMC 10A Problems/Problem 19|Solution]]
 ==Problem 20==
+Point <math>X</math> lies outside regular pentagon <math>ABCDE</math> so that <math>\triangle BXE</math> is an equilateral triangle, as shown below. What is the degree measure of acute angle <math>\angle CXD</math>?
-Let <math>S</math> be a subset of <math>\{1, 2, 3, \dots, 2024\}</math> such that the following two conditions hold:
+<asy>
+import graph; size(7cm);
+real labelscalefactor = 0.75; /* changes label-to-point distance */
+pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
+pen dotstyle = black; /* point style */
+real xmin = -1.089556028373145, xmax = 2.738320502950249, ymin = -1.3143096399343266, ymax = 1.2691431521185288; /* image dimensions */
+pen qqwuqq = rgb(0,0.39215686274509803,0);
-*If <math>x</math> and <math>y</math> are distinct elements of <math>S</math>, then <math>|x-y| > 2.</math>
+filldraw(arc((1.9562952014676112,0),0.25,168,192)--(1.9562952014676112,0)--cycle, mediumgrey);
+/* draw figures */
-*If <math>x</math> and <math>y</math> are distinct odd elements of <math>S</math>, then <math>|x-y| > 6.</math>
+draw((-0.8090169943749473,0.5877852522924731)--(0.30901699437494745,0.9510565162951535));
+draw((0.30901699437494745,0.9510565162951535)--(1,0));
-What is the maximum possible number of elements in <math>S</math>?
+draw((1,0)--(0.3090169943749473,-0.9510565162951536));
+draw((0.3090169943749473,-0.9510565162951536)--(-0.8090169943749475,-0.587785252292473));
+draw((-0.8090169943749475,-0.587785252292473)--(-0.8090169943749473,0.5877852522924731));
+draw((0.30901699437494745,0.9510565162951535)--(0.3090169943749473,-0.9510565162951536));
+draw((0.3090169943749473,-0.9510565162951536)--(1.9562952014676112,0));
+draw((1.9562952014676112,0)--(0.30901699437494745,0.9510565162951535));
+draw((-0.8090169943749473,0.5877852522924731)--(1.9562952014676112,0), dashed);
+draw((1.9562952014676112,0)--(-0.8090169943749475,-0.587785252292473), dashed);
+/* dots and labels */
+dot((1,0),linewidth(4pt) + dotstyle);
+label("$A$", (1.013592864312142,0), E * labelscalefactor);
+dot((0.30901699437494745,0.9510565162951535),linewidth(4pt) + dotstyle);
+label("$B$", (0.26,1), NE * labelscalefactor);
+dot((-0.8090169943749473,0.5877852522924731),linewidth(4pt) + dotstyle);
+label("$C$", (-0.82,0.64), NW * 0.25);
+dot((-0.8090169943749475,-0.587785252292473),linewidth(4pt) + dotstyle);
+label("$D$", (-0.82,-0.64), SW * 0.25);
+dot((0.3090169943749473,-0.9510565162951536),linewidth(4pt) + dotstyle);
+label("$E$", (0.26,-1), SE * labelscalefactor);
+dot((1.9562952014676112,0),linewidth(4pt) + dotstyle);
+label("$X$", (1.9705619971429902, 0), E * labelscalefactor);
+clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
+/* end of picture */
+</asy>
-<math>\textbf{(A) }436 \qquad \textbf{(B) }506 \qquad \textbf{(C) }608 \qquad \textbf{(D) }654 \qquad \textbf{(E) }675</math>
+<math>\textbf{(A)}~18^{\circ}\qquad\textbf{(B)}~19.5^{\circ}\qquad\textbf{(C)}~21^{\circ}\qquad\textbf{(D)}~22.5^{\circ}\qquad\textbf{(E)}~24^{\circ}</math>
 [[2024 AMC 10A Problems/Problem 20|Solution]]
 ==Problem 21==
+A fair six-sided die is repeatedly rolled until the same number is rolled twice in a row.  What is the probability that the last number rolled is equal to the first number rolled?
-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>?
+<math>\textbf{(A)}~\frac{17}{72} \qquad\textbf{(B)}~\frac{4}{15} \qquad\textbf{(C)}~\frac{5}{18} \qquad\textbf{(D)}~\frac{2}{7} \qquad\textbf{(E)}~\frac{3}{10}</math>
-<cmath> \begin{bmatrix} . & ? &.&.&. \ .&.&.&48&.\ 12&.&.&.&.\ .&.&16&.&.\ .&.&.&.&0\end{bmatrix}</cmath>
-<math>\textbf{(A)}~19\qquad\textbf{(B)}~24\qquad\textbf{(C)}~29\qquad\textbf{(D)}~34\qquad\textbf{(E)}~39</math>
 [[2024 AMC 10A Problems/Problem 21|Solution]]
 ==Problem 22==
+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>?
-Let <math>\mathcal K</math> be the kite formed by joining two right triangles with legs <math>1</math> and <math>\sqrt3</math> along a common hypotenuse. Eight copies of <math>\mathcal K</math> are used to form the polygon shown below. What is the area of triangle <math>\Delta ABC</math>? [[File:Screenshot_2024-11-08_3.23.29_PM.png]]
+<math>\textbf{(A)}~18\qquad\textbf{(B)}~44\qquad\textbf{(C)}~108\qquad\textbf{(D)}~141\qquad\textbf{(E)}~636</math>
-<math>\textbf{(A) }2+3\sqrt3\qquad\textbf{(B) }\dfrac92\sqrt3\qquad\textbf{(C) }\dfrac{10+8\sqrt3}{3}\qquad\textbf{(D) }8\qquad\textbf{(E) }5\sqrt3</math>
 [[2024 AMC 10A Problems/Problem 22|Solution]]
 ==Problem 23==
+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?
-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) }212 \qquad
-\textbf{(B) }247 \qquad
-\textbf{(C) }258 \qquad
-\textbf{(D) }276 \qquad
-\textbf{(E) }284 \qquad
-</math>
-[[2024 AMC 10A Problems/Problem 23|Solution]]
-==Problem 24==
-A bee is moving in three-dimensional space. A fair six-sided die with faces labeled <math>A^+, A^-, B^+, B^-, C^+,</math> and <math>C^-</math> is rolled. Suppose the bee occupies the point <math>(a,b,c).</math> If the die shows <math>A^+</math>, then the bee moves to the point <math>(a+1,b,c)</math> and if the die shows <math>A^-,</math> then the bee moves to the point <math>(a-1,b,c).</math> Analogous moves are made with the other four outcomes. Suppose the bee starts at the point <math>(0,0,0)</math> and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
-<math>\textbf{(A) }\frac{1}{54}\qquad\textbf{(B) }\frac{7}{54}\qquad\textbf{(C) }\frac{1}{6}\qquad\textbf{(D) }\frac{5}{18}\qquad\textbf{(E) }\frac{2}{5}</math>
-[[2024 AMC 10A Problems/Problem 24|Solution]]
-==Problem 25==
-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>
 size(6cm);
@@ Line 237: / Line 264: @@
 }}}
 </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 10A Problems/Problem 23|Solution]]
+==Problem 24==
+There exists a unique two-digit prime number <math>p</math> such that both <math>4^{28} - 15</math> and <math>3^{28} - 20</math> are divisible by <math>p</math>. What is the sum of the digits of <math>p</math>?
+<math>\textbf{(A)}~10\qquad\textbf{(B)}~11\qquad\textbf{(C)}~13\qquad\textbf{(D)}~14\qquad\textbf{(E)}~16</math>
+[[2024 AMC 10A Problems/Problem 24|Solution]]
+==Problem 25==
+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>?
+<math>\textbf{(A)}~180\qquad\textbf{(B)}~181\qquad\textbf{(C)}~182\qquad\textbf{(D)}~183\qquad\textbf{(E)}~184</math>
 [[2024 AMC 10A Problems/Problem 25|Solution]]