Difference between revisions of "2012 BMO Problems/Problem 4"
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==Problem== | ==Problem== | ||
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>. | 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== | ||
This is the same problem as the 2012 USAMO Problem 4. See https://artofproblemsolving.com/wiki/index.php/2012_USAMO_Problems/Problem_4. | This is the same problem as the 2012 USAMO Problem 4. See https://artofproblemsolving.com/wiki/index.php/2012_USAMO_Problems/Problem_4. | ||
Latest revision as of 16:44, 1 January 2024
Problem
Find all functions 
(where 
is the set of positive integers) such that 
for all positive integers 
and such that 
divides 
for all distinct positive integers 
, .
Solution
This is the same problem as the 2012 USAMO Problem 4. See https://artofproblemsolving.com/wiki/index.php/2012_USAMO_Problems/Problem_4.
