Here is one I liked:

Putnam 1963 A2: Let $f:\ \mathbb{N}\rightarrow\mathbb{N}$ be a strictly increasing multiplicative function. (that is it satisfies $f(mn)=f(m)f(n)$ for any relatively prime integers $m,n$.) If $f(2)=2$, prove that $f(n)=n$ for $n\in \mathbb{N}$.

The problem is fairly standard, and follows by induction. However, it was motivated by the following generalization due to Erdos:

If $f:\mathbb{N}\rightarrow \mathbb{R}$ is multiplicative and monotonically increasing, then $f(n)=n^\alpha$ for some $\alpha$.

A very

Two nice , short proof is given hereproofs of this generalization can be found in the American Mathematical Monthly. See this article by Everett Howe or this article by Joel Cohen.

Here is one I liked:

Putnam 1963 A2: Let $f:\ \mathbb{N}\rightarrow\mathbb{N}$ be a strictly increasing multiplicative function. (that is it satisfies $f(mn)=f(m)f(n)$ for any relatively prime integers $m,n$.) If $f(2)=2$, prove that $f(n)=n$ for $n\in \mathbb{N}$.

The problem is fairly standard, and follows by induction. However, it was motivated by the following generalization due to Erdos:

If $f:\mathbb{N}\rightarrow \mathbb{R}$ is multiplicative and monotonically increasing, then $f(n)=n^\alpha$ for some $\alpha$.

A very nice, short proof is given here by Joel Cohen.