Difference between revisions of "2024 AMC 10A Problems"

Latest revision as of 17:04, 23 November 2024

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 $9901\cdot101-99\cdot10101?$

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

Solution

Problem 2

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 $T=aL+bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimates it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?

$\textbf{(A) }240\qquad\textbf{(B) }246\qquad\textbf{(C) }252\qquad\textbf{(D) }258\qquad\textbf{(E) }264$

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?

$\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }13$

Solution

Problem 4

The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?

$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$

Solution

Problem 5

What is the least value of $n$ such that $n!$ is a multiple of $2024$ ?

$\textbf{(A) } 11\qquad\textbf{(B) } 21\qquad\textbf{(C) } 22\qquad\textbf{(D) } 23\qquad\textbf{(E) } 253$

Solution

Problem 6

What is the minimum number of successive swaps of adjacent letters in the string $ABCDEF$ that are needed to change the string to $FEDCBA?$ (For example, $3$ swaps are required to change $ABC$ to $CBA;$ one such sequence of swaps is $ABC\to BAC\to BCA\to CBA.$ )

$\textbf{(A)}~6\qquad\textbf{(B)}~10\qquad\textbf{(C)}~12\qquad\textbf{(D)}~15\qquad\textbf{(E)}~24$

Solution

Problem 7

The product of three integers is $60$ . What is the least possible positive sum of the three integers?

$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }13$

Solution

Problem 8

Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at $1:00 PM$ and were able to pack $4$ , $3$ , and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group, and Daria was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2:45 PM$ . At what time did Daria join the group?

$\textbf{(A) }1:25\text{ PM}\qquad\textbf{(B) }1:35\text{ PM}\qquad\textbf{(C) }1:45\text{ PM}\qquad\textbf{(D) }1:55\text{ PM}\qquad\textbf{(E) }2:05\text{ PM}$

Solution

Problem 9

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 10

Consider the following operation. Given a positive integer $n$ , if $n$ is a multiple of $3$ , then you replace $n$ by $\frac{n}{3}$ . If $n$ is not a multiple of $3$ , then you replace $n$ by $n+10$ . For example, beginning with $n=4$ , this procedure gives $4\to14\to24\to8\to18\to6\to2\to12\to\cdots$ . Suppose you start with $n=100$ . What value results if you perform this operation exactly $100$ times?

$\textbf{(A) }10\qquad\textbf{(B) }20\qquad\textbf{(C) }30\qquad\textbf{(D) }40\qquad\textbf{(E) }50$

Solution

Problem 11

How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$ ?

$\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}$ Infinitely many

Solution

Problem 12

Zelda played the Adventures of Math game on August 1 and scored $1,700$ points. She continued to play daily over the next $5$ 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 $1,700 + 80 = 1,780$ points.) What was Zelda's average score in points over the $6$ days?

$\textbf{(A)}~1700\qquad\textbf{(B)}~1702\qquad\textbf{(C)}~1703\qquad\textbf{(D)}~1713\qquad\textbf{(E)}~1715$

Solution

Problem 13

Two transformations are said to commute if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:

a translation $2$ units to the right,

a $90^{\circ}$ -rotation counterclockwise about the origin,

a reflection across the $x$ -axis, and

a dilation centered at the origin with scale factor $2.$

Of the $6$ pairs of distinct transformations from this list, how many commute?

$\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5$

Solution

Problem 14

One side of an equilateral triangle of height $24$ lies on line $\ell$ . A circle of radius $12$ is tangent to line $\ell$ 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 $\ell$ can be written as $a \sqrt{b} - c \pi$ , where $a$ , $b$ , and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is $a + b + c$ ?

$\textbf{(A)}~72\qquad\textbf{(B)}~73\qquad\textbf{(C)}~74\qquad\textbf{(D)}~75\qquad\textbf{(E)}~76$

Solution

Problem 15

Let $M$ be the greatest integer such that both $M+1213$ and $M+3773$ are perfect squares. What is the units digit of $M$ ?

$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

Solution

Problem 16

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 $AB$ ? $\newline$ $\textbf{(A) }4+4\sqrt5\qquad\textbf{(B) }10\sqrt2\qquad\textbf{(C) }5+5\sqrt5\qquad\textbf{(D) }10\sqrt[4]{8}\qquad\textbf{(E) }20$

Solution

Problem 17

Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ 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 $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$ . Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$ . Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$ , where $m$ and $n$ are positive integers. What is $m+n$ ?

$\textbf{(A)}~10\qquad\textbf{(B)}~11\qquad\textbf{(C)}~12\qquad\textbf{(D)}~13\qquad\textbf{(E)}~14$

Solution

Problem 18

There are exactly $K$ positive integers $5 \leq b \leq 2024$ such that the base- $b$ integer $2024_{b}$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$ ?

$\textbf{(A)}~16\qquad\textbf{(B)}~17\qquad\textbf{(C)}~18\qquad\textbf{(D)}~20\qquad\textbf{(E)}~21$

Solution

Problem 19

The first three terms of a geometric sequence are the integers $a, 720$ and $b$ , where $a < 720 < b$ . What is the sum of the digits of the least possible value of $b$ ?

$\textbf{(A) } 9\qquad\textbf{(B) } 12\qquad\textbf{(C) } 16\qquad\textbf{(D) } 18\qquad\textbf{(E) } 21$

Solution

Problem 20

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

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

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

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

Solution

Problem 21

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 22

Let $\mathcal K$ be the kite formed by joining two right triangles with legs $1$ and $\sqrt3$ along a common hypotenuse. Eight copies of $\mathcal K$ are used to form the polygon shown below. What is the area of triangle $\Delta ABC$ ?

$\textbf{(A) }2+3\sqrt3\qquad\textbf{(B) }\dfrac92\sqrt3\qquad\textbf{(C) }\dfrac{10+8\sqrt3}{3}\qquad\textbf{(D) }8\qquad\textbf{(E) }5\sqrt3$

Solution

Problem 23

Integers $a$ , $b$ , and $c$ satisfy $ab + c = 100$ , $bc + a = 87$ , and $ca + b = 60$ . What is $ab + bc + ca?$

$\textbf{(A) }212 \qquad \textbf{(B) }247 \qquad \textbf{(C) }258 \qquad \textbf{(D) }276 \qquad \textbf{(E) }284 \qquad$

Solution

Problem 24

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 25

The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ 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

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by 2023 AMC 10B Problems	Followed by 2024 AMC 10B Problems
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
All AMC 10 Problems and Solutions

Page

Toolbox

Search

Difference between revisions of "2024 AMC 10A Problems"

Latest revision as of 17:04, 23 November 2024

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 3: / Line 3: @@
 ==Problem 1==
-Cities <math>A</math> and <math>B</math> are <math>45</math> miles apart. Alicia lives in <math>A</math> and Beth lives in <math>B</math>. Alicia bikes towards <math>B</math> at <math>18</math> miles per hour. Leaving at the same time, Beth bikes toward <math>A</math> at <math>12</math> miles per hour. How many miles from City <math>A</math> will they be when they meet?
+What is the value of <math>9901\cdot101-99\cdot10101?</math>
-<math>\textbf{(A) }20\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad\textbf{(E) }27</math>
+<math>\textbf{(A)}~2\qquad\textbf{(B)}~20\qquad\textbf{(C)}~200\qquad\textbf{(D)}~202\qquad\textbf{(E)}~2020</math>
-[[2023 AMC 10A Problems/Problem 1|Solution]]
+[[2024 AMC 10A Problems/Problem 1|Solution]]
 ==Problem 2==
-The weight of <math>\frac{1}{3}</math> of a large pizza together with <math>3 \frac{1}{2}</math> cups of orange slices is the same weight of <math>\frac{3}{4}</math> of a large pizza together with <math>\frac{1}{2}</math> cups of orange slices. A cup of orange slices weigh <math>\frac{1}{4}</math> of a pound. What is the weight, in pounds, of a large pizza?
+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\frac{4}{5}\qquad\textbf{(B) }2\qquad\textbf{(C) }2\frac{2}{5}\qquad\textbf{(D) }3\qquad\textbf{(E) }3\frac{3}{5}</math>
+<math>\textbf{(A) }240\qquad\textbf{(B) }246\qquad\textbf{(C) }252\qquad\textbf{(D) }258\qquad\textbf{(E) }264</math>
-[[2023 AMC 10A Problems/Problem 2|Solution]]
+[[2024 AMC 10A Problems/Problem 2|Solution]]
 ==Problem 3==
-How many positive perfect squares less than <math>2023</math> are divisible by <math>5</math>?
+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) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12</math>
+<math>\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }13</math>
-[[2023 AMC 10A Problems/Problem 3|Solution]]
+[[2024 AMC 10A Problems/Problem 3|Solution]]
 ==Problem 4==
-A quadrilateral has all integer side lengths, a perimeter of <math>26</math>, and one side of length <math>4</math>. What is the greatest possible length of one side of this quadrilateral?
+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?
-<math>\textbf{(A) }9\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }12\qquad\textbf{(E) }13</math>
+<math>\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24</math>
-[[2023 AMC 10A Problems/Problem 4|Solution]]
+[[2024 AMC 10A Problems/Problem 4|Solution]]
 ==Problem 5==
-How many digits are in the base-ten representation of <math>8^5 \cdot 5^{10} \cdot 15^5</math>?
+What is the least value of <math>n</math> such that <math>n!</math> is a multiple of <math>2024</math>?
-<math>\textbf{(A) } 14 \qquad\textbf{(B) }15 \qquad\textbf{(C) }16 \qquad\textbf{(D) }17 \qquad\textbf{(E) } 18</math>
+<math>\textbf{(A) } 11\qquad\textbf{(B) } 21\qquad\textbf{(C) } 22\qquad\textbf{(D) } 23\qquad\textbf{(E) } 253</math>
-[[2023 AMC 10A Problems/Problem 5|Solution]]
+[[2024 AMC 10A Problems/Problem 5|Solution]]
 ==Problem 6==
-An integer is assigned to each vertex of a cube. The value of an edge is defined to be the sum of the values of the two vertices it touches, and the value of a face is defined to be the sum of the values of the four edges surrounding it. The value of the cube is defined as the sum of the values of its six faces. Suppose the sum of the integers assigned to the vertices is <math>21</math>. What is the value of the cube?
+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>ABC\to BAC\to BCA\to CBA.</math>)
-<math>\textbf{(A) } 42 \qquad \textbf{(B) } 63 \qquad \textbf{(C) } 84 \qquad \textbf{(D) } 126 \qquad \textbf{(E) } 252</math>
+<math>\textbf{(A)}~6\qquad\textbf{(B)}~10\qquad\textbf{(C)}~12\qquad\textbf{(D)}~15\qquad\textbf{(E)}~24</math>
-[[2023 AMC 10A Problems/Problem 6|Solution]]
+[[2024 AMC 10A Problems/Problem 6|Solution]]
 ==Problem 7==
-Janet rolls a standard <math>6</math>-sided die <math>4</math> times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal <math>3?</math>
+The product of three integers is <math>60</math>. What is the least possible positive sum of the
+three integers?
-<math>\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}</math>
+<math>\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }13</math>
-[[2023 AMC 10A Problems/Problem 7|Solution]]
+[[2024 AMC 10A Problems/Problem 7|Solution]]
 ==Problem 8==
-Barb the baker has developed a new temperature scale for her bakery called the Breadus scale, which is a linear function of the Fahrenheit scale. Bread rises at <math>110</math> degrees Fahrenheit, which is <math>0</math> degrees on the Breadus scale. Bread is baked at <math>350</math> degrees Fahrenheit, which is <math>100</math> degrees on the Breadus scale. Bread is done when its internal temperature is <math>200</math> degrees Fahrenheit. What is this, in degrees, on the Breadus scale?
+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) }33\qquad\textbf{(B) }34.5\qquad\textbf{(C) }36\qquad\textbf{(D) }37.5\qquad\textbf{(E) }39</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>
-[[2023 AMC 10A Problems/Problem 8|Solution]]
+[[2024 AMC 10A Problems/Problem 8|Solution]]
 ==Problem 9==
-A digital display shows the current date as an <math>8</math>-digit integer consisting of a <math>4</math>-digit year, followed by a <math>2</math>-digit month, followed by a <math>2</math>-digit date within the month. For example, Arbor Day this year is displayed as <math>20230428.</math> For how many dates in <math>2023</math> does each digit appear an even number of times in the <math>8</math>-digital display for that date?
+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?
-<math>\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9</math>
+<math>\textbf{(A) }720\qquad\textbf{(B) }1350\qquad\textbf{(C) }2700\qquad\textbf{(D) }3280\qquad\textbf{(E) }8100</math>
-[[2023 AMC 10A Problems/Problem 9|Solution]]
+[[2024 AMC 10A Problems/Problem 9|Solution]]
 ==Problem 10==
-Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an <math>11</math> on the next quiz, her mean will increase by <math>1</math>. If she scores an <math>11</math> on each of the next three quizzes, her mean will increase by <math>2</math>. What is the mean of her quiz scores currently?
+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) }10\qquad\textbf{(B) }20\qquad\textbf{(C) }30\qquad\textbf{(D) }40\qquad\textbf{(E) }50</math>
-<math>\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8</math>
+[[2024 AMC 10A Problems/Problem 10|Solution]]
-[[2023 AMC 10A Problems/Problem 10|Solution]]
 ==Problem 11==
-A square of area <math>2</math> is inscribed in a square of area <math>3</math>, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
+How many ordered pairs of integers <math>(m, n)</math> satisfy <math>\sqrt{n^2 - 49} = m</math>?
-<asy>
-size(200);
-defaultpen(linewidth(0.6pt)+fontsize(10pt));
-real y = sqrt(3);
-pair A,B,C,D,E,F,G,H;
-A = (0,0);
-B = (0,y);
-C = (y,y);
-D = (y,0);
-E = ((y + 1)/2,y);
-F = (y, (y - 1)/2);
-G = ((y - 1)/2, 0);
-H = (0,(y + 1)/2);
-fill(H--B--E--cycle, gray);
-draw(A--B--C--D--cycle);
-draw(E--F--G--H--cycle);
-</asy>
-<math>\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1</math>
+<math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}</math> Infinitely many
-[[2023 AMC 10A Problems/Problem 11|Solution]]
+[[2024 AMC 10A Problems/Problem 11|Solution]]
 ==Problem 12==
-How many three-digit positive integers <math>N</math> satisfy the following properties?
+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)}~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]]
-* The number <math>N</math> is divisible by <math>7</math>.
+==Problem 13==
+Two transformations are said to commute if applying the first followed by the second
+gives the same result as applying the second followed by the first. Consider these
+four transformations of the coordinate plane:
-* The number formed by reversing the digits of <math>N</math> is divisible by <math>5</math>.
+* a translation <math>2</math> units to the right,
-<math>\textbf{(A) } 13 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17</math>
+* a <math>90^{\circ}</math>-rotation counterclockwise about the origin,
-[[2023 AMC 10A Problems/Problem 12|Solution]]
+* a reflection across the <math>x</math>-axis, and
-==Problem 13==
+* a dilation centered at the origin with scale factor <math>2.</math>
-Abdul and Chiang are standing <math>48</math> feet apart in a field. Bharat is standing in the same field as far from Abdul as possible so that the angle formed by his lines of sight to Abdul and Chiang measures <math>60^\circ</math>. What is the square of the distance (in feet) between Abdul and Bharat?
+Of the <math>6</math> pairs of distinct transformations from this list, how many commute?
-<math>\textbf{(A) } 1728 \qquad \textbf{(B) } 2601 \qquad \textbf{(C) } 3072 \qquad \textbf{(D) } 4608 \qquad \textbf{(E) } 6912</math>
+<math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}~5</math>
-[[2023 AMC 10A Problems/Problem 13|Solution]]
+[[2024 AMC 10A Problems/Problem 13|Solution]]
 ==Problem 14==
-A number is chosen at random from among the first <math>100</math> positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by <math>11</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>?
-<math>\textbf{(A)}~\frac{4}{100}\qquad\textbf{(B)}~\frac{9}{200} \qquad \textbf{(C)}~\frac{1}{20} \qquad\textbf{(D)}~\frac{11}{200}\qquad\textbf{(E)}~\frac{3}{50}</math>
+<math>\textbf{(A)}~72\qquad\textbf{(B)}~73\qquad\textbf{(C)}~74\qquad\textbf{(D)}~75\qquad\textbf{(E)}~76</math>
-[[2023 AMC 10A Problems/Problem 14|Solution]]
+[[2024 AMC 10A Problems/Problem 14|Solution]]
 ==Problem 15==
-An even number of circles are nested, starting with a radius of <math>1</math> and increasing by <math>1</math> each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius <math>2</math> but outside the circle of radius <math>1.</math> An example showing <math>8</math> circles is displayed below. What is the least number of circles needed to make the total shaded area at least <math>2023\pi</math>?
-<asy>
+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>?
-size(6cm);
-pen greywhat;
-greywhat = RGB(105,105,105);
-filldraw(circle((8, 0), 8), greywhat);
-filldraw(circle((7, 0), 7), white);
-filldraw(circle((6, 0), 6), greywhat);
-filldraw(circle((5, 0), 5), white);
-filldraw(circle((4, 0), 4), greywhat);
-filldraw(circle((3, 0), 3), white);
-filldraw(circle((2, 0), 2), greywhat);
-filldraw(circle((1, 0), 1), white);
-</asy>
-<math>\textbf{(A) } 46 \qquad \textbf{(B) } 48 \qquad \textbf{(C) } 56 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 64</math>
+<math>\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8</math>
-[[2023 AMC 10A Problems/Problem 15|Solution]]
+[[2024 AMC 10A Problems/Problem 15|Solution]]
 ==Problem 16==
+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>
+[[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>
-In a table tennis tournament every participant played every other participant exactly once. Although there were twice as many right-handed players as left-handed players, the number of games won by left-handed players was <math>40\%</math> more than the number of games won by right-handed players. (There were no ties and no ambidextrous players.) What is the total number of games played?
+[[2024 AMC 10A Problems/Problem 16|Solution]]
-<math>\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66</math>
+==Problem 17==
-[[2023 AMC 10A Problems/Problem 16|Solution]]
+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>?
-==Problem 17==
+<math>\textbf{(A)}~10\qquad\textbf{(B)}~11\qquad\textbf{(C)}~12\qquad\textbf{(D)}~13\qquad\textbf{(E)}~14</math>
-Let <math>ABCD</math> be a rectangle with <math>AB = 30</math> and <math>BC = 28</math>. Point <math>P</math> and <math>Q</math> lie on <math>\overline{BC}</math> and <math>\overline{CD}</math> respectively so that all sides of <math>\triangle{ABP}, \triangle{PCQ},</math> and <math>\triangle{QDA}</math> have integer lengths. What is the perimeter of <math>\triangle{APQ}</math>?
-<math>\textbf{(A) } 84 \qquad \textbf{(B) } 86 \qquad \textbf{(C) } 88 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } 92</math>
+[[2024 AMC 10A Problems/Problem 17|Solution]]
-[[2023 AMC 10A Problems/Problem 17|Solution]]
+==Problem 18==
-==Problem 18==
+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>?
-A rhombic dodecahedron is a solid with <math>12</math> congruent rhombus faces. At every vertex, <math>3</math> or <math>4</math> edges meet, depending on the vertex. How many vertices have exactly <math>3</math> edges meet?
-<math>\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9</math>
+<math>\textbf{(A)}~16\qquad\textbf{(B)}~17\qquad\textbf{(C)}~18\qquad\textbf{(D)}~20\qquad\textbf{(E)}~21</math>
-[[2023 AMC 10A Problems/Problem 18|Solution]]
+[[2024 AMC 10A Problems/Problem 18|Solution]]
 ==Problem 19==
-The line segment formed by <math>A(1, 2)</math> and <math>B(3, 3)</math> is rotated to the line segment formed by <math>A'(3, 1)</math> and <math>B'(4, 3)</math> about the point <math>P(r, s)</math>. What is <math>|r-s|</math>?
-<math>\textbf{(A) } \frac{1}{4} \qquad \textbf{(B) } \frac{1}{2} \qquad \textbf{(C) } \frac{3}{4}   \qquad \textbf{(D) } \frac{2}{3} \qquad   \textbf{(E) } 1</math>
+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>?
-[[2023 AMC 10A Problems/Problem 19|Solution]]
+<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==
-Each square in a <math>3\times3</math> grid of squares is colored red, white, blue, or green so that every <math>2\times2</math> square contains one square of each color. One such coloring is shown on the right below. How many different colorings are possible?
-<asy>
+Let <math>S</math> be a subset of <math>\{1, 2, 3, \dots, 2024\}</math> such that the following two conditions hold: <math>\linebreak</math>
-size(8cm);
+*If <math>x</math> and <math>y</math> are distinct elements of <math>S</math>, then <math>|x-y| > 2.</math>   <math>\newline</math>
-pen grey1, grey2, grey3;
+*If <math>x</math> and <math>y</math> are distinct odd elements of <math>S</math>, then <math>|x-y| > 6.</math> <math>\newline</math>
-grey1 = RGB(211, 211, 211);
+What is the maximum possible number of elements in <math>S</math>?
-grey2 = RGB(173, 173, 173);
-grey3 = RGB(138, 138, 138);
-for(int i = 0; i < 4; ++i) {
+<math>\textbf{(A) }436 \qquad \textbf{(B) }506 \qquad \textbf{(C) }608 \qquad \textbf{(D) }654 \qquad \textbf{(E) }675 \qquad</math>
-draw((i, 0)--(i, 3));
-draw((0, i)--(3, i));
-}
-filldraw((5, 3)--(6, 3)--(6, 2)--(5, 2)--cycle, grey1);
+[[2024 AMC 10A Problems/Problem 20|Solution]]
-label('B', (5.5, 2.5));
-filldraw((6, 3)--(7, 3)--(7, 2)--(6, 2)--cycle, grey2);
-label('R', (6.5, 2.5));
-filldraw((7, 3)--(8, 3)--(8, 2)--(7, 2)--cycle, grey1);
-label('B', (7.5, 2.5));
-filldraw((5, 2)--(6, 2)--(6, 1)--(5, 1)--cycle, grey3);
-label('G', (5.5, 1.5));
-filldraw((6, 2)--(7, 2)--(7, 1)--(6, 1)--cycle, white);
-label('W', (6.5, 1.5));
-filldraw((7, 2)--(8, 2)--(8, 1)--(7, 1)--cycle, grey3);
-label('G', (7.5, 1.5));
-filldraw((5, 1)--(6, 1)--(6, 0)--(5, 0)--cycle, grey2);
-label('R', (5.5, 0.5));
-filldraw((6, 1)--(7, 1)--(7, 0)--(6, 0)--cycle, grey1);
-label('B', (6.5, 0.5));
-filldraw((7, 1)--(8, 1)--(8, 0)--(7, 0)--cycle, grey2);
-label('R', (7.5, 0.5));
-</asy>
-<math>\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }96</math>
+==Problem 21==
-[[2023 AMC 10A Problems/Problem 20|Solution]]
+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>
-==Problem 21==
-Let <math>P(x)</math> be the unique polynomial of minimal degree with the following properties:
-*<math>P(x)</math> has a leading coefficient <math>1</math>,
+<math>\textbf{(A)}~19\qquad\textbf{(B)}~24\qquad\textbf{(C)}~29\qquad\textbf{(D)}~34\qquad\textbf{(E)}~39</math>
-*<math>1</math> is a root of <math>P(x)-1</math>,
+[[2024 AMC 10A Problems/Problem 21|Solution]]
-*<math>2</math> is a root of <math>P(x-2)</math>,
+==Problem 22==
-*<math>3</math> is a root of <math>P(3x)</math>, and
+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>4</math> is a root of <math>4P(x)</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>
-The roots of <math>P(x)</math> are integers, with one exception. The root that is not an integer can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime integers. What is <math>m+n</math>?
+[[2024 AMC 10A Problems/Problem 22|Solution]]
-<math>\textbf{(A) }41\qquad\textbf{(B) }43\qquad\textbf{(C) }45\qquad\textbf{(D) }47\qquad\textbf{(E) }49</math>
+==Problem 23==
-[[2023 AMC 10A Problems/Problem 21|Solution]]
+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>
-==Problem 22==
+<math>
-Circle <math>C_1</math> and <math>C_2</math> each have radius <math>1</math>, and the distance between their centers is <math>\frac{1}{2}</math>. Circle <math>C_3</math> is the largest circle internally tangent to both <math>C_1</math> and <math>C_2</math>. Circle <math>C_4</math> is internally tangent to both <math>C_1</math> and <math>C_2</math> and externally tangent to <math>C_3</math>. What is the radius of <math>C_4</math>?
+\textbf{(A) }212 \qquad
+\textbf{(B) }247 \qquad
+\textbf{(C) }258 \qquad
+\textbf{(D) }276 \qquad
+\textbf{(E) }284 \qquad
+</math>
-<asy>
+[[2024 AMC 10A Problems/Problem 23|Solution]]
-import olympiad;
-size(10cm);
-draw(circle((0,0),0.75));
-draw(circle((-0.25,0),1));
-draw(circle((0.25,0),1));
-draw(circle((0,6/7),3/28));
-pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118);
-dot(B^^C);
-draw(B--E, dashed);
-draw(C--F, dashed);
-draw(B--C);
-label("$C_4$", D);
-label("$C_1$", (-1.375, 0));
-label("$C_2$", (1.375,0));
-label("$\frac{1}{2}$", (0, -.125));
-label("$C_3$", (-0.4, -0.4));
-label("$1$", (-.85, 0.70));
-label("$1$", (.85, -.7));
-import olympiad;
-markscalefactor=0.005;
-</asy>
-<math>\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}</math>
+==Problem 24==
-[[2023 AMC 10A Problems/Problem 22|Solution]]
+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?
-==Problem 23==
+<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>
-If the positive integer <math>c</math> has positive integer divisors <math>a</math> and <math>b</math> with <math>c = ab</math>, then <math>a</math> and <math>b</math> are said to be <math>\textit{complementary}</math> divisors of <math>c</math>. Suppose that <math>N</math> is a positive integer that has one complementary pair of divisors that differ by <math>20</math> and another pair of complementary divisors that differ by <math>23</math>. What is the sum of the digits of <math>N</math>?
-<math>\textbf{(A) } 9 \qquad \textbf{(B) } 13\qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19</math>
+[[2024 AMC 10A Problems/Problem 24|Solution]]
-[[2023 AMC 10A Problems/Problem 23|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?
-==Problem 24==
-Six regular hexagonal blocks of side length <math>1</math> unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is <math>\frac{3}{7}</math> unit. What is the area of the region inside the frame not occupied by the blocks?
 <asy>
-unitsize(1cm);
+size(6cm);
-draw(scale(3)*polygon(6));
+for (int i=0; i<9; ++i) {
-filldraw(shift(dir(0)*2+dir(120)*0.4)*polygon(6), lightgray);
+  draw((i,0)--(i,3),dotted);
-filldraw(shift(dir(60)*2+dir(180)*0.4)*polygon(6), lightgray);
+}
-filldraw(shift(dir(120)*2+dir(240)*0.4)*polygon(6), lightgray);
+for (int i=0; i<4; ++i){
-filldraw(shift(dir(180)*2+dir(300)*0.4)*polygon(6), lightgray);
+  draw((0,i)--(8,i),dotted);
-filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray);
+}
-filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray);
+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>
-<math>\textbf{(A)}~\frac{13 \sqrt{3}}{3}\qquad\textbf{(B)}~\frac{216 \sqrt{3}}{49}\qquad\textbf{(C)}~\frac{9 \sqrt{3}}{2} \qquad\textbf{(D)}~ \frac{14 \sqrt{3}}{3}\qquad\textbf{(E)}~\frac{243 \sqrt{3}}{49}</math>
+<math>\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }162\qquad\textbf{(E) }196</math>
-[[2023 AMC 10A Problems/Problem 24|Solution]]
-==Problem 25==
-If <math>A</math> and <math>B</math> are vertices of a polyhedron, define the distance <math>d(A, B)</math> to be the minimum number of edges of the polyhedron one must traverse in order to connect <math>A</math> and <math>B</math>. For example, <math>\overline{AB}</math> is an edge of the polyhedron, then <math>d(A, B) = 1</math>, but if <math>\overline{AC}</math> and <math>\overline{CB}</math> are edges and <math>\overline{AB}</math> is not an edge, then <math>d(A, B) = 2</math>. Let <math>Q</math>, <math>R</math>, and <math>S</math> be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of <math>20</math> equilateral triangles). What is the probability that <math>d(Q, R) > d(R, S)</math>?
-<math>\textbf{(A) }\frac{7}{22}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{3}{8}\qquad\textbf{(D) }\frac{5}{12}\qquad\textbf{(E) }\frac{1}{2}</math>
-[[2023 AMC 10A Problems/Problem 25|Solution]]
+[[2024 AMC 10A Problems/Problem 25|Solution]]
 ==See also==