Difference between revisions of "2012 USAMO Problems/Problem 4"

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(Problem)
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== Problem ==
 
== Problem ==
  
Give 370 proofs of the Pythagorean theorem.
+
Find all functions <math>f : \mathbb{Z}^+ \to \mathbb{Z}^+</math> (where <math>\mathbb{Z}^+</math> is the set of positive integers) such that <math>f(n!) = f(n)!</math> for all positive integers <math>n</math> and such that <math>m - n</math> divides <math>f(m) - f(n)</math> for all distinct positive integers <math>m</math>, <math>n</math>.
  
 
==Solution==
 
==Solution==

Revision as of 17:54, 25 April 2012

Problem

Find all functions $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m - n$ divides $f(m) - f(n)$ for all distinct positive integers $m$, $n$.

Solution

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