Difference between revisions of "2011 IMO Problems/Problem 5"

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Let f be a function from the set of integers to the set of positive integers. Suppose that, for any two integers m and n, the difference f (m) f (n) is divisible by f (m n). Prove that, for all integers m and n with f(m) f(n), the number f(n) is divisible by f(m).
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Let <math>f</math> be a function from the set of integers to the set of positive integers. Suppose that, for any two integers <math>m</math> and <math>n</math>, the difference <math>f(m) - f(n)</math> is divisible by <math>f(m - n)</math>. Prove that, for all integers <math>m</math> and <math>n</math> with <math>f(m) \leq f(n)</math>, the number <math>f(n)</math> is divisible by <math>f(m)</math>.

Revision as of 13:41, 27 November 2011

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m - n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$.

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