# 1969 Canadian MO Problems/Problem 8

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## Problem

Let $f$ be a function with the following properties:

1) $f(n)$ is defined for every positive integer $n$;

2) $f(n)$ is an integer;

3) $f(2)=2$;

4) $f(mn)=f(m)f(n)$ for all $m$ and $n$;

5) $f(m)>f(n)$ whenever $m>n$.

Prove that $f(n)=n$.

## Solution

It's easily shown that $f(1)=1$ and $f(4)=4$. Since $f(2) $f(3) = 3.$

Now, assume that $f(2n+2)=f(2(n+1))=f(2)f(n+1)=2n+2$ is true for all $f(k)$ where $k\leq 2n.$

It follows that $2n Hence, $f(2n+1)=2n+1$, and by induction $f(n) = n$.

 1969 Canadian MO (Problems) Preceded byProblem 7 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • Followed byProblem 9