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Difference between revisions of "2024 AMC 10A Problems"
Line 107: Line 107:
 
==Problem 14==
 
==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>?
+
XXX
  
−
<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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
  
−
[[2023 AMC 10A Problems/Problem 14|Solution]]
+
[[2024 AMC 10A Problems/Problem 14|Solution]]
  
 
==Problem 15==
 
==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>
+
XXX
−
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) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
  
−
[[2023 AMC 10A Problems/Problem 15|Solution]]
+
[[2024 AMC 10A Problems/Problem 15|Solution]]
  
 
==Problem 16==
 
==Problem 16==
  
 +
XXX
  
−
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?
+
<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
  
−
<math>\textbf{(A) }15\qquad\textbf{(B) }36\qquad\textbf{(C) }45\qquad\textbf{(D) }48\qquad\textbf{(E) }66</math>
+
[[2024 AMC 10A Problems/Problem 16|Solution]]
  
−
[[2023 AMC 10A Problems/Problem 16|Solution]]
+
==Problem 17==
  
−
==Problem 17==
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XXX
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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>
+
<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
  
−
[[2023 AMC 10A Problems/Problem 17|Solution]]
+
[[2024 AMC 10A Problems/Problem 17|Solution]]
  
 
==Problem 18==
 
==Problem 18==
−
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?
 
  
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<math>\textbf{(A) }5\qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9</math>
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XXX
 +
 
 +
<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
  
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[[2023 AMC 10A Problems/Problem 18|Solution]]
+
[[2024 AMC 10A Problems/Problem 18|Solution]]
  
 
==Problem 19==
 
==Problem 19==
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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>?
 
  
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<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>
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XXX
 +
 
 +
<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
  
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[[2023 AMC 10A Problems/Problem 19|Solution]]
+
[[2024 AMC 10A Problems/Problem 19|Solution]]
  
 
==Problem 20==
 
==Problem 20==
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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?
 
−
 
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<asy>
 
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size(8cm);
 
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pen grey1, grey2, grey3;
 
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grey1 = RGB(211, 211, 211);
 
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grey2 = RGB(173, 173, 173);
 
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grey3 = RGB(138, 138, 138);
 
−
 
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for(int i = 0; i < 4; ++i) {
 
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draw((i, 0)--(i, 3));
 
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draw((0, i)--(3, i));
 
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}
 
  
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filldraw((5, 3)--(6, 3)--(6, 2)--(5, 2)--cycle, grey1);
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XXX
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label('B', (5.5, 2.5));
 
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filldraw((6, 3)--(7, 3)--(7, 2)--(6, 2)--cycle, grey2);
 
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label('R', (6.5, 2.5));
 
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filldraw((7, 3)--(8, 3)--(8, 2)--(7, 2)--cycle, grey1);
 
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label('B', (7.5, 2.5));
 
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filldraw((5, 2)--(6, 2)--(6, 1)--(5, 1)--cycle, grey3);
 
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label('G', (5.5, 1.5));
 
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filldraw((6, 2)--(7, 2)--(7, 1)--(6, 1)--cycle, white);
 
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label('W', (6.5, 1.5));
 
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filldraw((7, 2)--(8, 2)--(8, 1)--(7, 1)--cycle, grey3);
 
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label('G', (7.5, 1.5));
 
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filldraw((5, 1)--(6, 1)--(6, 0)--(5, 0)--cycle, grey2);
 
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label('R', (5.5, 0.5));
 
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filldraw((6, 1)--(7, 1)--(7, 0)--(6, 0)--cycle, grey1);
 
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label('B', (6.5, 0.5));
 
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filldraw((7, 1)--(8, 1)--(8, 0)--(7, 0)--cycle, grey2);
 
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label('R', (7.5, 0.5));
 
−
</asy>
 
  
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<math>\textbf{(A) }24\qquad\textbf{(B) }48\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }96</math>
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<math>\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }</math>
  
−
[[2023 AMC 10A Problems/Problem 20|Solution]]
+
[[2024 AMC 10A Problems/Problem 20|Solution]]
  
 
==Problem 21==
 
==Problem 21==

Revision as of 15:04, 8 November 2024

2024 AMC 10A (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. 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.
  2. 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.
  3. 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).
  4. Figures are not necessarily drawn to scale.
  5. 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

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 2

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 3

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 4

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 5

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 6

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 7

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 8

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 9

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 10

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 11

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 12

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 13

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 14

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 15

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 16

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 17

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 18

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 19

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 20

XXX

$\textbf{(A) }\qquad\textbf{(B) }\qquad\textbf{(C) }\qquad\textbf{(D) }\qquad\textbf{(E) }$

Solution

Problem 21

Let $P(x)$ be the unique polynomial of minimal degree with the following properties:

  • $P(x)$ has a leading coefficient $1$,
  • $1$ is a root of $P(x)-1$,
  • $2$ is a root of $P(x-2)$,
  • $3$ is a root of $P(3x)$, and
  • $4$ is a root of $4P(x)$.

The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. What is $m+n$?

$\textbf{(A) }41\qquad\textbf{(B) }43\qquad\textbf{(C) }45\qquad\textbf{(D) }47\qquad\textbf{(E) }49$

Solution

Problem 22

Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$?

[asy] 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]

$\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}$

Solution

Problem 23

If the positive integer $c$ has positive integer divisors $a$ and $b$ with $c = ab$, then $a$ and $b$ are said to be $\textit{complementary}$ divisors of $c$. Suppose that $N$ is a positive integer that has one complementary pair of divisors that differ by $20$ and another pair of complementary divisors that differ by $23$. What is the sum of the digits of $N$?

$\textbf{(A) } 9 \qquad \textbf{(B) } 13\qquad \textbf{(C) } 15 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19$

Solution

Problem 24

Six regular hexagonal blocks of side length $1$ 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 $\frac{3}{7}$ unit. What is the area of the region inside the frame not occupied by the blocks? [asy] unitsize(1cm); draw(scale(3)*polygon(6)); filldraw(shift(dir(0)*2+dir(120)*0.4)*polygon(6), lightgray); filldraw(shift(dir(60)*2+dir(180)*0.4)*polygon(6), lightgray); filldraw(shift(dir(120)*2+dir(240)*0.4)*polygon(6), lightgray); filldraw(shift(dir(180)*2+dir(300)*0.4)*polygon(6), lightgray); filldraw(shift(dir(240)*2+dir(360)*0.4)*polygon(6), lightgray); filldraw(shift(dir(300)*2+dir(420)*0.4)*polygon(6), lightgray); [/asy] $\textbf{(A)}~\frac{13 \sqrt{3}}{3}\qquad\textbf{(B)}~\frac{216 \sqrt{3}}{49}\qquad\textbf{(C)}~\frac{9 \sqrt{3}}{2} \qquad\textbf{(D)}~ \frac{14 \sqrt{3}}{3}\qquad\textbf{(E)}~\frac{243 \sqrt{3}}{49}$

Solution

Problem 25

If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of $20$ equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?

$\textbf{(A) }\frac{7}{22}\qquad\textbf{(B) }\frac{1}{3}\qquad\textbf{(C) }\frac{3}{8}\qquad\textbf{(D) }\frac{5}{12}\qquad\textbf{(E) }\frac{1}{2}$

Solution

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
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