Difference between revisions of "2025 USAMO Problems/Problem 5"
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Determine, with proof, all positive integers <math>k</math> such that<cmath>\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k</cmath>is an integer for every positive integer <math>n.</math> | Determine, with proof, all positive integers <math>k</math> such that<cmath>\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k</cmath>is an integer for every positive integer <math>n.</math> | ||
− | == Solution == | + | == Solution 1== |
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+ | https://artofproblemsolving.com/wiki/index.php/File:2025_USAMO_PROBLEM_5_1.jpg | ||
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+ | https://artofproblemsolving.com/wiki/index.php/File:2025_USAMO_PROBLEM_5_2.jpg | ||
==See Also== | ==See Also== | ||
{{USAMO newbox|year=2025|num-b=4|num-a=6}} | {{USAMO newbox|year=2025|num-b=4|num-a=6}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 08:30, 10 April 2025
Contents
[hide]Problem
Determine, with proof, all positive integers such that
is an integer for every positive integer
Solution 1
https://artofproblemsolving.com/wiki/index.php/File:2025_USAMO_PROBLEM_5_1.jpg
https://artofproblemsolving.com/wiki/index.php/File:2025_USAMO_PROBLEM_5_2.jpg
See Also
2025 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.