Difference between revisions of "2024 AMC 10A Problems"
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{{AMC10 Problems|year=2024|ab=A}} | {{AMC10 Problems|year=2024|ab=A}} | ||
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==Problem 1== | ==Problem 1== | ||
− | + | What is the value of <math>9901\cdot101-99\cdot10101?</math> | |
− | <math> \textbf{(A)} | + | <math>\textbf{(A)}~2\qquad\textbf{(B)}~20\qquad\textbf{(C)}~200\qquad\textbf{(D)}~202\qquad\textbf{(E)}~2020</math> |
+ | |||
+ | [[2024 AMC 10A Problems/Problem 1|Solution]] | ||
==Problem 2== | ==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 <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) }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== | ==Problem 3== | ||
− | |||
− | <math>\textbf{(A)} | + | 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> | ||
+ | |||
+ | [[2024 AMC 10A Problems/Problem 3|Solution]] | ||
==Problem 4== | ==Problem 4== | ||
− | + | ||
− | + | 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)} | + | |
− | \textbf{(B)} | + | <math>\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24</math> |
− | \textbf{(C)} | + | |
− | \textbf{(D)} | + | [[2024 AMC 10A Problems/Problem 4|Solution]] |
− | \textbf{(E)} | ||
==Problem 5== | ==Problem 5== | ||
− | |||
− | <math>\textbf{(A) } | + | What is the least value of <math>n</math> such that <math>n!</math> is a multiple of <math>2024</math>? |
− | \textbf{(B) } | + | |
− | \textbf{(C) } | + | <math>\textbf{(A) } 11\qquad\textbf{(B) } 21\qquad\textbf{(C) } 22\qquad\textbf{(D) } 23\qquad\textbf{(E) } 253</math> |
− | \textbf{(D) } | + | |
− | \textbf{(E) } | + | [[2024 AMC 10A Problems/Problem 5|Solution]] |
==Problem 6== | ==Problem 6== | ||
− | + | 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)}~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== | ==Problem 7== | ||
− | + | The product of three integers is <math>60</math>. What is the least possible positive sum of the | |
+ | three integers? | ||
− | <math> | + | <math>\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }13</math> |
− | \textbf{(A) } | + | |
− | \textbf{(B) } | + | [[2024 AMC 10A Problems/Problem 7|Solution]] |
− | \textbf{(C) } | ||
− | \textbf{(D) } | ||
− | \textbf{(E) } | ||
− | </math> | ||
==Problem 8== | ==Problem 8== | ||
− | On | + | 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) }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== | ==Problem 9== | ||
− | In | + | 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) }720\qquad\textbf{(B) }1350\qquad\textbf{(C) }2700\qquad\textbf{(D) }3280\qquad\textbf{(E) }8100</math> |
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− | } | ||
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− | } | ||
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− | |||
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− | } | ||
− | </ | ||
− | + | [[2024 AMC 10A Problems/Problem 9|Solution]] | |
==Problem 10== | ==Problem 10== | ||
− | |||
− | <math>\ | + | 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> | ||
+ | |||
+ | [[2024 AMC 10A Problems/Problem 10|Solution]] | ||
==Problem 11== | ==Problem 11== | ||
− | |||
− | <math>\textbf{(A)} | + | How many ordered pairs of integers <math>(m, n)</math> satisfy <math>\sqrt{n^2 - 49} = m</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]] | ||
==Problem 12== | ==Problem 12== | ||
− | + | 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)} | + | <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== | ==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 <math>2</math> units to the right, | |
− | <math> \textbf{(A)} | + | * 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]] | ||
==Problem 14== | ==Problem 14== | ||
− | The | + | 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)}~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]] | ||
+ | |||
+ | ==Problem 15== | ||
− | <math> | + | Let <math>M</math> be the greatest integer such that both <math>M+1213</math> and <math>M+3773</math> are perfect squares. What is the units digit of <math>M</math>? |
− | + | <math>\textbf{(A) }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]] | |
==Problem 16== | ==Problem 16== | ||
− | All the | + | 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> | ||
− | + | [[2024 AMC 10A Problems/Problem 16|Solution]] | |
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==Problem 17== | ==Problem 17== | ||
− | |||
− | <math>\ | + | 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)}~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== | ==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>? | |
− | <math> \textbf{(A)} | + | <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]] | ||
==Problem 19== | ==Problem 19== | ||
− | + | 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) } 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== | ==Problem 20== | ||
− | |||
− | <math>\textbf{(A) } | + | Let <math>S</math> be a subset of <math>\{1, 2, 3, \dots, 2024\}</math> such that the following two conditions hold: |
+ | |||
+ | *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> | ||
+ | |||
+ | 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 20|Solution]] | ||
==Problem 21== | ==Problem 21== | ||
− | + | 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)} | + | |
+ | <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== | ==Problem 22== | ||
− | |||
− | <math>\textbf{(A) } | + | 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) }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== | ==Problem 23== | ||
− | |||
− | <math>\textbf{(A)} | + | 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== | ==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== | ==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); | ||
+ | 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> | ||
+ | <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 25|Solution]] | |
− | |||
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==See also== | ==See also== | ||
Line 192: | Line 247: | ||
* [[Mathematics competitions]] | * [[Mathematics competitions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
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Latest revision as of 05:53, 14 December 2024
2024 AMC 10A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
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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 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
What is the value of
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 where and are constants, is the time in minutes, is the length of the trail in miles, and is the altitude gain in feet. The model estimates that it will take minutes to hike to the top if a trail is miles long and ascends feet, as well as if a trail is miles long and ascends feet. How many minutes does the model estimates it will take to hike to the top if the trail is miles long and ascends feet?
Problem 3
What is the sum of the digits of the smallest prime that can be written as a sum of distinct primes?
Problem 4
The number 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?
Problem 5
What is the least value of such that is a multiple of ?
Problem 6
What is the minimum number of successive swaps of adjacent letters in the string that are needed to change the string to (For example, swaps are required to change to one such sequence of swaps is )
Problem 7
The product of three integers is . What is the least possible positive sum of the three integers?
Problem 8
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at and were able to pack , , and packages, respectively, every minutes. At some later time, Daria joined the group, and Daria was able to pack packages every minutes. Together, they finished packing packages at exactly . At what time did Daria join the group?
Problem 9
In how many ways can juniors and seniors form disjoint teams of people so that each team has juniors and seniors?
Problem 10
Consider the following operation. Given a positive integer , if is a multiple of , then you replace by . If is not a multiple of , then you replace by . For example, beginning with , this procedure gives . Suppose you start with . What value results if you perform this operation exactly times?
Problem 11
How many ordered pairs of integers satisfy ?
Infinitely many
Problem 12
Zelda played the Adventures of Math game on August 1 and scored points. She continued to play daily over the next 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 points.) What was Zelda's average score in points over the days?
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 units to the right,
- a -rotation counterclockwise about the origin,
- a reflection across the -axis, and
- a dilation centered at the origin with scale factor
Of the pairs of distinct transformations from this list, how many commute?
Problem 14
One side of an equilateral triangle of height lies on line . A circle of radius is tangent to line 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 can be written as , where , , and are positive integers and is not divisible by the square of any prime. What is ?
Problem 15
Let be the greatest integer such that both and are perfect squares. What is the units digit of ?
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 ?
Problem 17
Two teams are in a best-two-out-of-three playoff: the teams will play at most games, and the winner of the playoff is the first team to win 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 chance of winning at home, and its probability of winning when playing away from home is . Outcomes of the games are independent. The probability that Team A wins the playoff is . Then can be written in the form , where and are positive integers. What is ?
Problem 18
There are exactly positive integers such that the base- integer is divisible by (where is in base ten). What is the sum of the digits of ?
Problem 19
The first three terms of a geometric sequence are the integers and , where . What is the sum of the digits of the least possible value of ?
Problem 20
Let be a subset of such that the following two conditions hold:
- If and are distinct elements of , then
- If and are distinct odd elements of , then
What is the maximum possible number of elements in ?
Problem 21
The numbers, in order, of each row and the numbers, in order, of each column of a array of integers form an arithmetic progression of length . The numbers in positions , , and are , , , and , respectively. What number is in position ?
Problem 22
Let be the kite formed by joining two right triangles with legs and along a common hypotenuse. Eight copies of are used to form the polygon shown below. What is the area of triangle ?
Problem 23
Integers , , and satisfy , , and . What is
Problem 24
A bee is moving in three-dimensional space. A fair six-sided die with faces labeled and is rolled. Suppose the bee occupies the point If the die shows , then the bee moves to the point and if the die shows then the bee moves to the point Analogous moves are made with the other four outcomes. Suppose the bee starts at the point and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
Problem 25
The figure below shows a dotted grid cells wide and cells tall consisting of squares. Carl places -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?
See also
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 |