Difference between revisions of "2024 AMC 10A Problems"
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==Problem 4== | ==Problem 4== | ||
+ | What is <math>\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?</math> | ||
+ | |||
+ | <math>\textbf{(A)}\ -1 \qquad | ||
+ | \textbf{(B)}\ \frac{5}{36} \qquad | ||
+ | \textbf{(C)}\ \frac{7}{12} \qquad | ||
+ | \textbf{(D)}\ \frac{147}{60} \qquad | ||
+ | \textbf{(E)}\ \frac{43}{3} </math> | ||
==Problem 5== | ==Problem 5== | ||
+ | At the theater children get in for half price. The price for <math>5</math> adult tickets and <math>4</math> child tickets is <math>\$24.50</math>. How much would <math>8</math> adult tickets and <math>6</math> child tickets cost? | ||
+ | |||
+ | <math>\textbf{(A) }\$35\qquad | ||
+ | \textbf{(B) }\$38.50\qquad | ||
+ | \textbf{(C) }\$40\qquad | ||
+ | \textbf{(D) }\$42\qquad | ||
+ | \textbf{(E) }\$42.50</math> | ||
==Problem 6== | ==Problem 6== | ||
+ | |||
+ | The area of a pizza with radius <math>4</math> is <math>N</math> percent larger than the area of a pizza with radius <math>3</math> inches. What is the integer closest to <math>N</math>? | ||
+ | |||
+ | <math>\textbf{(A) } 25 \qquad\textbf{(B) } 33 \qquad\textbf{(C) } 44\qquad\textbf{(D) } 66 \qquad\textbf{(E) } 78</math> | ||
==Problem 7== | ==Problem 7== | ||
+ | |||
+ | A circle has a chord of length <math>10</math>, and the distance from the center of the circle to the chord is <math>5</math>. What is the area of the circle? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A) }25\pi \qquad | ||
+ | \textbf{(B) }50\pi \qquad | ||
+ | \textbf{(C) }75\pi \qquad | ||
+ | \textbf{(D) }100\pi \qquad | ||
+ | \textbf{(E) }125\pi \qquad | ||
+ | </math> | ||
==Problem 8== | ==Problem 8== | ||
+ | |||
+ | On an algebra quiz, <math>10\%</math> of the students scored <math>70</math> points, <math>35\%</math> scored <math>80</math> points, <math>30\%</math> scored <math>90</math> points, and the rest scored <math>100</math> points. What is the difference between the mean and median score of the students' scores on this quiz? | ||
+ | |||
+ | <math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math> | ||
==Problem 9== | ==Problem 9== | ||
+ | |||
+ | In the plane figure shown below, <math>3</math> of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry? | ||
+ | |||
+ | <asy> | ||
+ | import olympiad; | ||
+ | unitsize(25); | ||
+ | filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); | ||
+ | filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); | ||
+ | filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); | ||
+ | for (int i = 0; i < 5; ++i) { | ||
+ | for (int j = 0; j < 6; ++j) { | ||
+ | pair A = (j,i); | ||
+ | } | ||
+ | } | ||
+ | for (int i = 0; i < 5; ++i) { | ||
+ | for (int j = 0; j < 6; ++j) { | ||
+ | if (j != 5) { | ||
+ | draw((j,i)--(j+1,i)); | ||
+ | } | ||
+ | if (i != 4) { | ||
+ | draw((j,i)--(j,i+1)); | ||
+ | } | ||
+ | } | ||
+ | } | ||
+ | </asy> | ||
+ | |||
+ | <math>\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8</math> | ||
==Problem 10== | ==Problem 10== | ||
+ | The functions <math>\sin(x)</math> and <math>\cos(x)</math> are periodic with least period <math>2\pi</math>. What is the least period of the function <math>\cos(\sin(x))</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{\pi}{2}\qquad\textbf{(B)}\ \pi\qquad\textbf{(C)}\ 2\pi \qquad\textbf{(D)}\ 4\pi \qquad\textbf{(E)} </math> The function is not periodic. | ||
==Problem 11== | ==Problem 11== | ||
+ | Let <math>x</math> and <math>y</math> be two-digit positive integers with mean <math>60</math>. What is the maximum value of the ratio <math>\frac{x}{y}</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}</math> | ||
==Problem 12== | ==Problem 12== | ||
+ | |||
+ | A frog sitting at the point <math>(1, 2)</math> begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length <math>1</math>, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices <math>(0,0), (0,4), (4,4),</math> and <math>(4,0)</math>. What is the probability that the sequence of jumps ends on a vertical side of the square? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac12\qquad\textbf{(B)}\ \frac 58\qquad\textbf{(C)}\ \frac 23\qquad\textbf{(D)}\ \frac34\qquad\textbf{(E)}\ \frac 78</math> | ||
==Problem 13== | ==Problem 13== | ||
+ | |||
+ | What is the minimum number of digits to the right of the decimal point needed to express the fraction <math>\frac{123456789}{2^{26}\cdot 5^4}</math> as a decimal? | ||
+ | |||
+ | <math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 22\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 104</math> | ||
==Problem 14== | ==Problem 14== | ||
+ | |||
+ | The sequence | ||
+ | |||
+ | <math>\log_{12}{162}</math>, <math>\log_{12}{x}</math>, <math>\log_{12}{y}</math>, <math>\log_{12}{z}</math>, <math>\log_{12}{1250}</math> | ||
+ | |||
+ | is an arithmetic progression. What is <math>x</math>? | ||
+ | |||
+ | <math> \textbf{(A)} \ 125\sqrt{3} \qquad \textbf{(B)} \ 270 \qquad \textbf{(C)} \ 162\sqrt{5} \qquad \textbf{(D)} \ 434 \qquad \textbf{(E)} \ 225\sqrt{6}</math> | ||
==Problem 16== | ==Problem 16== | ||
+ | |||
+ | All the numbers <math>2, 3, 4, 5, 6, 7</math> are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products? | ||
+ | |||
+ | <math>\textbf{(A)}\ 312 \qquad | ||
+ | \textbf{(B)}\ 343 \qquad | ||
+ | \textbf{(C)}\ 625 \qquad | ||
+ | \textbf{(D)}\ 729 \qquad | ||
+ | \textbf{(E)}\ 1680</math> | ||
==Problem 17== | ==Problem 17== | ||
+ | Jesse cuts a circular disk of radius 12, along 2 radii to form 2 sectors, one with a central angle of 120. He makes two circular cones using each sector to form the lateral surface of each cone. What is the ratio of the volume of the smaller cone to the larger cone? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac{1}{8}\qquad\textbf{(B)}\ \frac{1}{4}\qquad\textbf{(C)}\ \frac{\sqrt{10}}{10}\qquad\textbf{(D)}\ \frac{\sqrt{5}}{6}\qquad\textbf{(E)}\ \frac{\sqrt{5}}{5}</math> | ||
==Problem 18== | ==Problem 18== | ||
+ | |||
+ | Rhombus <math>ABCD</math> has side length <math>2</math> and <math>\angle B = 120</math>°. Region <math>R</math> consists of all points inside the rhombus that are closer to vertex <math>B</math> than any of the other three vertices. What is the area of <math>R</math>? | ||
+ | |||
+ | <math> \textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad\textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad\textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad\textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad\textbf{(E)}\ 2</math> | ||
+ | [[Category: Introductory Geometry Problems]] | ||
==Problem 19== | ==Problem 19== | ||
+ | |||
+ | Let <math>p</math> and <math>q</math> be positive integers such that <cmath>\frac{5}{9} < \frac{p}{q} < \frac{4}{7}</cmath>and <math>q</math> is as small as possible. What is <math>q-p</math>? | ||
+ | |||
+ | <math>\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 </math> | ||
==Problem 20== | ==Problem 20== | ||
+ | There exists a unique strictly increasing sequence of nonnegative integers <math>a_1 < a_2 < … < a_k</math> such that<cmath>\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.</cmath>What is <math>k?</math> | ||
+ | |||
+ | <math>\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306</math> | ||
==Problem 21== | ==Problem 21== | ||
+ | |||
+ | In <math>\triangle{ABC}</math> with side lengths <math>AB = 13</math>, <math>AC = 12</math>, and <math>BC = 5</math>, let <math>O</math> and <math>I</math> denote the circumcenter and incenter, respectively. A circle with center <math>M</math> is tangent to the legs <math>AC</math> and <math>BC</math> and to the circumcircle of <math>\triangle{ABC}</math>. What is the area of <math>\triangle{MOI}</math>? | ||
+ | |||
+ | <math>\textbf{(A)}\ \frac52\qquad\textbf{(B)}\ \frac{11}{4}\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ \frac{13}{4}\qquad\textbf{(E)}\ \frac72</math> | ||
==Problem 22== | ==Problem 22== | ||
+ | A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is <math>3\sqrt3</math> inches, its top diameter is <math>6</math> inches, and its bottom diameter is <math>12</math> inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey? | ||
+ | |||
+ | <math>\textbf{(A) } 6 + 3\pi\qquad \textbf{(B) }6 + 6\pi\qquad \textbf{(C) } 6\sqrt3 \qquad \textbf{(D) } 6\sqrt5 \qquad \textbf{(E) } 6\sqrt3 + \pi</math> | ||
==Problem 23== | ==Problem 23== | ||
+ | Let <math>f</math> be the unique function defined on the positive integers such that <cmath>\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1</cmath> for all positive integers <math>n</math>. What is <math>f(2023)</math>? | ||
+ | |||
+ | <math>\textbf{(A)}~-1536\qquad\textbf{(B)}~96\qquad\textbf{(C)}~108\qquad\textbf{(D)}~116\qquad\textbf{(E)}~144</math> | ||
==Problem 24== | ==Problem 24== | ||
+ | Let <math>a</math>, <math>b</math>, and <math>c</math> be positive integers with <math>a\ge</math> <math>b\ge</math> <math>c</math> such that | ||
+ | <math>a^2-b^2-c^2+ab=2011</math> and | ||
+ | <math>a^2+3b^2+3c^2-3ab-2ac-2bc=-1997</math>. | ||
+ | |||
+ | What is <math>a</math>? | ||
+ | |||
+ | <math> \textbf{(A)}\ 249\qquad\textbf{(B)}\ 250\qquad\textbf{(C)}\ 251\qquad\textbf{(D)}\ 252\qquad\textbf{(E)}\ 253 </math> | ||
==Problem 25== | ==Problem 25== | ||
− | |||
− | + | A rectangular box measures <math>a \times b \times c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers and <math>1\leq a \leq b \leq c</math>. The volume and the surface area of the box are numerically equal. How many ordered triples <math>(a,b,c)</math> are possible? | |
+ | |||
+ | <math>\textbf{(A)}\; 4 \qquad\textbf{(B)}\; 10 \qquad\textbf{(C)}\; 12 \qquad\textbf{(D)}\; 21 \qquad\textbf{(E)}\; 26</math> | ||
==See also== | ==See also== |
Latest revision as of 23:35, 9 September 2024
2024 AMC 10A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
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 16
- 16 Problem 17
- 17 Problem 18
- 18 Problem 19
- 19 Problem 20
- 20 Problem 21
- 21 Problem 22
- 22 Problem 23
- 23 Problem 24
- 24 Problem 25
- 25 See also
Problem 1
A bug crawls along a number line, starting at . It crawls to
, then turns around and crawls to
. How many units does the bug crawl altogether?
Problem 2
What is the value of ?
Problem 3
When counting from to
,
is the
number counted. When counting backwards from
to
,
is the
number counted. What is
?
Problem 4
What is
Problem 5
At the theater children get in for half price. The price for adult tickets and
child tickets is
. How much would
adult tickets and
child tickets cost?
Problem 6
The area of a pizza with radius is
percent larger than the area of a pizza with radius
inches. What is the integer closest to
?
Problem 7
A circle has a chord of length , and the distance from the center of the circle to the chord is
. What is the area of the circle?
Problem 8
On an algebra quiz, of the students scored
points,
scored
points,
scored
points, and the rest scored
points. What is the difference between the mean and median score of the students' scores on this quiz?
Problem 9
In the plane figure shown below, of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?
Problem 10
The functions and
are periodic with least period
. What is the least period of the function
?
The function is not periodic.
Problem 11
Let and
be two-digit positive integers with mean
. What is the maximum value of the ratio
?
Problem 12
A frog sitting at the point begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length
, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices
and
. What is the probability that the sequence of jumps ends on a vertical side of the square?
Problem 13
What is the minimum number of digits to the right of the decimal point needed to express the fraction as a decimal?
Problem 14
The sequence
,
,
,
,
is an arithmetic progression. What is ?
Problem 16
All the numbers are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?
Problem 17
Jesse cuts a circular disk of radius 12, along 2 radii to form 2 sectors, one with a central angle of 120. He makes two circular cones using each sector to form the lateral surface of each cone. What is the ratio of the volume of the smaller cone to the larger cone?
Problem 18
Rhombus has side length
and
°. Region
consists of all points inside the rhombus that are closer to vertex
than any of the other three vertices. What is the area of
?
Problem 19
Let and
be positive integers such that
and
is as small as possible. What is
?
Problem 20
There exists a unique strictly increasing sequence of nonnegative integers such that
What is
Problem 21
In with side lengths
,
, and
, let
and
denote the circumcenter and incenter, respectively. A circle with center
is tangent to the legs
and
and to the circumcircle of
. What is the area of
?
Problem 22
A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is inches, its top diameter is
inches, and its bottom diameter is
inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. What is the length in inches of its shortest path to the honey?
Problem 23
Let be the unique function defined on the positive integers such that
for all positive integers
. What is
?
Problem 24
Let ,
, and
be positive integers with
such that
and
.
What is ?
Problem 25
A rectangular box measures , where
,
, and
are integers and
. The volume and the surface area of the box are numerically equal. How many ordered triples
are possible?
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.