Here’s a nice problem that was among some IMO practice problems. I don’t know the source.
Problem: Prove that for each natural number $n$, there is some natural number $r$ for which the $n$ integers $r+1^2,r+2^2,\ldots r+n^2$ are all squarefree.
Solution (sketch): For a large prime $p$, the probability that none of these $n$ integers is divisible by $p^2$ is $1-\frac{n}{p^2}$. Assuming independence for the $p_i$s we get $(1-\frac{n}{2^2})(1-\frac{n}{3^2})(1-\frac{n}{5^2})\ldots$, which converges to a positive number, so there must exist solutions.
To make this into a rigorous argument one needs CRT.
