Difference between revisions of "2024 AMC 10A Problems"
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==Problem 21== | ==Problem 21== | ||
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+ | 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>? | ||
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+ | <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== |
Revision as of 22:33, 9 September 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
[hide]- 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 thatWhat 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
Problem 23
Problem 24
Problem 25
Stop trying to cheat!
~ TRX74x94Planet9
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.