Difference between revisions of "2024 AIME II Problems"
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==Problem 13== | ==Problem 13== | ||
− | + | Let <math>\omega\neq 1</math> be a 13th root of unity. Find the remainder when | |
− | + | <cmath> | |
+ | \prod_{k=0}^{12}(2-2\omega^k+\omega^{2k}) | ||
+ | </cmath> | ||
+ | is divided by 1000. | ||
[[2024 AIME II Problems/Problem 13|Solution]] | [[2024 AIME II Problems/Problem 13|Solution]] |
Revision as of 21:49, 8 February 2024
2024 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Among the residents of Aimeville, there are who own a diamond ring, who own a set of golf clubs, and who own a garden spade. In addition, each of the residents owns a bag of candy hearts. There are residents who own exactly two of these things, and residents who own exactly three of these things. Find the number of residents of Aimeville who own all four of these things.
Problem 2
A list of positive integers has the following properties:
The sum of the items in the list is .
The unique mode of the list is .
The median of the list is a positive integer that does not appear in the list itself.
Find the sum of the squares of all the items in the list.
Problem 3
Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is , and the sum of the three numbers formed by reading top to bottom is . The grid below is an example of such an arrangement because and .
Problem 4
Problem 5
Problem 6
Problem 7
Let be the greatest four-digit integer with the property that whenever one of its digits is changed to , the resulting number is divisible by . Let and be the quotient and remainder, respectively, when is divided by . Find .
Problem 8
Problem 9
Problem 10
Problem 11
Find the number of triples of nonnegative integers satisfying and
Problem 12
Problem 13
Let be a 13th root of unity. Find the remainder when is divided by 1000.
Problem 14
Problem 15
See also
2024 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2024 AIME I |
Followed by 2025 AIME I | |
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.