Difference between revisions of "2023 AIME I Problems"
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==Problem 2== | ==Problem 2== | ||
− | + | Positive real numbers <math>b \not= 1</math> and <math>n</math> satisfy the equations <cmath>\sqrt{\log_b n} = \log_b \sqrt{n} \qquad \text{and} \qquad b \cdot \log_b n = \log_b (bn).</cmath> The value of <math>n</math> is <math>\frac{j}{k},</math> where <math>j</math> and <math>k</math> are relatively prime positive integers. Find <math>j+k.</math> | |
[[2023 AIME I Problems/Problem 2|Solution]] | [[2023 AIME I Problems/Problem 2|Solution]] |
Revision as of 13:24, 8 February 2023
2023 AIME I (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
Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is where and are relatively prime positive integers. Find
Problem 2
Positive real numbers and satisfy the equations The value of is where and are relatively prime positive integers. Find
Problem 3
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.
Problem 4
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 5
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.
Problem 6
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 7
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.
Problem 8
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 9
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.
Problem 10
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 11
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.
Problem 12
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 13
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 14
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 15
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
See also
2023 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2022 AIME II |
Followed by 2023 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.