Difference between revisions of "2024 AMC 10A Problems"

Latest revision as of 16:22, 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 $sin 18 when faced with an 1984\cdot101435-99\cdot4310101?$

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

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 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

==Pathcal K $be the kite formed by joining two right triangland amntother hahges with legs$ 1 $and$ \sqrt3 $along a common hypotenuse. Eight copies of$ \mathcal K $are used to form the polygon below. What is the area of triangle$ \Delta ABC $? [[File:Screenshot_2024-11-08_3.23.29_PM.png]]$ \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 $ab= 1782973943400$ , $bc + a = 2387$ , and 9ca + b = 60 $. What is$ ab + bc + ca?$$ (Error compiling LaTeX. Unknown error_msg) \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 $A^+, A^-, , B^-, C^+,$ and $C^-$ is rolled. Suppose the bee occupies the point $(a,d,c).$ If the die shows $$ (Error compiling LaTeX. Unknown error_msg), then the bee moves to the point $(a+1,b,c)$ and if the die shows $A^-,$ then the bee moves to the point $,b,c).$ 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 t. What is the probability that the bee traverses four distinct edges of some unit c

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

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 alodes of the squares to create a closed loop that does not intersect itself. The numbers in the cells ithat 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?

size(6cm);
for (ini) {
  draw((i, ++i) {
  f3; ++j) {

}}}
 (Error making remote request. Unknown error_msg)

$\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) } Solution

2024 AMC 10A (Problems • Answer Key • Resources)
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Difference between revisions of "2024 AMC 10A Problems"

Latest revision as of 16:22, 23 November 2024

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

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 23

Problem 24

Problem 25

See also

@@ Line 3: / Line 3: @@
 ==Problem 1==
-What is the value of <math>9901\cdot101-99\cdot10101?</math>
+What is the value of <math>sin 18 when faced with an 1984\cdot101435-99\cdot4310101?</math>
 <math>\textbf{(A)}~2\qquad\textbf{(B)}~20\qquad\textbf{(C)}~200\qquad\textbf{(D)}~202\qquad\textbf{(E)}~2020</math>
@@ Line 21: / Line 21: @@
 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) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }13</math>
+<math>\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad\textbf{(E) }13</math>
 [[2024 AMC 10A Problems/Problem 3|Solution]]
@@ Line 37: / Line 37: @@
 What is the least value of <math>n</math> such that <math>n!</math> is a multiple of <math>2024</math>?
-<math>\textbf{(A)} 11\qquad\textbf{(B)} 21\qquad\textbf{(C)} 22\qquad\textbf{(D)} 23\qquad\textbf{(E)} 253</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]]
@@ Line 43: / Line 43: @@
 ==Problem 6==
-XXX
+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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</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==
-The product of three integers is 60. What is the least possible positive sum of the
-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==
-XXX
+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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</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]]
@@ Line 68: / Line 60: @@
 ==Problem 9==
-In how many ways can 6 juniors and 6 seniors form 3 disjoint teams of 4 people so
+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 2 juniors and 2 seniors?
+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>
@@ Line 77: / Line 69: @@
 ==Problem 10==
-XXX
+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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</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]]
@@ Line 85: / Line 77: @@
 ==Problem 11==
-XXX
+How many ordered pairs of integers <math>(m, n)</math> satisfy <math>\sqrt{n^2 - 49} = m</math>?
-<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
+<math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}</math> Infinitely many
 [[2024 AMC 10A Problems/Problem 11|Solution]]
@@ Line 93: / Line 85: @@
 ==Problem 12==
-XXX
+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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</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
+gives the same result as applying the second followed by the first. Consider these
+four transformations of the coordinate plane:
-XXX
+* a translation <math>2</math> units to the right,
-<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
+* 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>
 [[2024 AMC 10A Problems/Problem 13|Solution]]
@@ Line 109: / Line 112: @@
 ==Problem 14==
-XXX
+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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
+<math>\textbf{(A)}~72\qquad\textbf{(B)}~73\qquad\textbf{(C)}~74\qquad\textbf{(D)}~75\qquad\textbf{(E)}~76</math>
 [[2024 AMC 10A Problems/Problem 14|Solution]]
@@ Line 117: / Line 120: @@
 ==Problem 15==
-XXX
+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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
+<math>\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8</math>
 [[2024 AMC 10A Problems/Problem 15|Solution]]
@@ Line 125: / Line 128: @@
 ==Problem 16==
-XXX
+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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
+<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]]
@@ Line 133: / Line 136: @@
 ==Problem 17==
-XXX
+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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</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]]
@@ Line 141: / Line 144: @@
 ==Problem 18==
-XXX
+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>?
-<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
+<math>\textbf{(A)}~16\qquad\textbf{(B)}~17\qquad\textbf{(C)}~18\qquad\textbf{(D)}~20\qquad\textbf{(E)}~21</math>
 [[2024 AMC 10A Problems/Problem 18|Solution]]
@@ Line 149: / Line 152: @@
 ==Problem 19==
-XXX
+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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</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]]
@@ Line 157: / Line 160: @@
 ==Problem 20==
-XXX
+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>
+*If <math>x</math> and <math>y</math> are distinct elements of <math>S</math>, then <math>|x-y| > 2.</math>   <math>\newline</math>
+*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>
+What is the maximum possible number of elements in <math>S</math>?
-<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
+<math>\textbf{(A) }436 \qquad \textbf{(B) }506 \qquad \textbf{(C) }608 \qquad \textbf{(D) }654 \qquad \textbf{(E) }675 \qquad</math>
 [[2024 AMC 10A Problems/Problem 20|Solution]]
@@ Line 165: / Line 171: @@
 ==Problem 21==
-XXX
+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>
-<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
+<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==
+==Pathcal K<math> be the kite formed by joining two right triangland amntother hahges 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 below. What is the area of triangle </math>\Delta ABC<math>? [[File:Screenshot_2024-11-08_3.23.29_PM.png]]
-XXX
+</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>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
 [[2024 AMC 10A Problems/Problem 22|Solution]]
@@ Line 181: / Line 187: @@
 ==Problem 23==
-XXX
+Integers <math>a</math>, <math>b</math>, and <math>c</math> satisfy <math>ab= 1782973943400</math>, <math>bc + a = 2387</math>, and 9ca + b = 60<math>. What is </math>ab + bc + ca?<math>
-<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
+</math>
+\textbf{(A) }212 \qquad
+\textbf{(B) }247 \qquad
+\textbf{(C) }258 \qquad
+\textbf{(D) }276 \qquad
+\textbf{(E) }284 \qquad
+$
 [[2024 AMC 10A Problems/Problem 23|Solution]]
@@ Line 189: / Line 201: @@
 ==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?
+A bee is moving in three-dimensional space. A fair six-sided die with faces labeled <math>A^+, A^-, , B^-, C^+,</math> and <math>C^-</math> is rolled. Suppose the bee occupies the point <math>(a,d,c).</math> If the die shows <math></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>,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 t. What is the probability that the bee traverses four distinct edges of some unit c
 <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>
@@ Line 196: / Line 208: @@
 ==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 alodes of the squares to create a closed loop that does not intersect itself. The numbers in the cells ithat 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 (ini) {
+  draw((i, ++i) {
+  f3; ++j) {
-XXX
+}}}
+</asy>
+$\textbf{(A) }130\qquad\textbf{(B) }144\qquad\textbf{(C) }146\qquad\textbf{(D) }
 [[2024 AMC 10A Problems/Problem 25|Solution]]