2024 AIME I Problems
2024 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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Contents
Problem 1
Every morning, Aya does a kilometer walk, and then finishes at the coffee shop. One day, she walks at kilometers per hour, and the walk takes hours, including minutes at the coffee shop. Another morning, she walks at kilometers per hour, and the walk takes hours and minutes, including minutes at the coffee shop. This morning, if she walks at kilometers per hour, how many minutes will the walk take, including the minutes at the coffee shop?
Problem 2
Real numbers and with satisfy What is the value of ?
Problem 3
Alice and Bob play the following game. A stack of tokens lies before them. The players take turns with Alice going first. On each turn, the player removes token or tokens from the stack. The player who removes the last token wins. Find the number of positive integers less than or equal to such that there is a strategy that guarantees that Bob wins, regardless of Alice’s moves.
Problem 4
Jen enters a lottery by picking distinct numbers from numbers are randomly chosen from She wins a prize if at least two of her numbers were of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of her winning the grand prize given that she won a prize is where and are relatively prime positive integers. Find .
Problem 5
Rectangles and are drawn such that are collinear. Also, all lie on a circle. If and what is the length of ?
Problem 6
Consider the paths of length that follow the lines from the lower left corner to the upper right corner on an grid. Find the number of such paths that change direction exactly four times, like in the examples shown below.
Problem 7
Find the largest possible real part of where is a complex number with .
Problem 8
Eight circles of radius can be placed tangent to of so that the circles are sequentially tangent to each other, with the first circle being tangent to and the last circle being tangent to , as shown. Similarly, circles of radius can be placed tangent to in the same manner. The inradius of can be expressed as , where and are relatively prime positive integers. Find .
Problem 9
Let be a rhombus whose vertices all lie on the hyperbola and are in that order. If its diagonals intersect at the origin, find the largest number less than for all rhombuses .
Problem 10
Let be a triangle inscribed in circle . Let the tangents to at and intersect at point , and let intersect at . If , , and , can be written as the form , where and are relatively prime integers. Find .
Problem 11
Each vertex of a regular octagon is independently colored either red or blue with equal probability. The probability that the octagon can then be rotated so that all of the blue vertices end up at positions where there were originally red vertices is , where and are relatively prime positive integers. What is ?
Problem 12
Define and . Find the number of intersections of the graphs of
Problem 13
Let be the least prime number for which there exists a positive integer such that is divisible by . Find the least positive integer such that is divisible by .
Problem 14
Let be a tetrahedron such that , , and . There exists a point inside the tetrahedron such that the distances from to each of the faces of the tetrahedron are all equal. This distance can be written in the form , when , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .
Problem 15
Let be the set of rectangular boxes with surface area and volume . Let be the radius of the smallest sphere that can contain each of the rectangular boxes that are elements of . The value of can be written as , where and are relatively prime positive integers. Find .
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2023 AIME II |
Followed by 2024 AIME II | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.