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 ==
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== Solution 1==
{{solution}}
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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

Problem

Determine, with proof, all positive integers $k$ such that\[\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k\]is an integer for every positive integer $n.$

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 (ProblemsResources)
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. AMC logo.png