Problem: Prove that there exists a real constant $c$
such that for any pair $(x,y)$ of real numbers,
there exist relatively prime integers $m$ and $n$
satisfying the relation
\[ \sqrt{(x-m)^2 + (y-n)^2} < c \log(x^2+y^2+2). \]

Solution: This is actually the same problem as USAMO 2014/6. Surprise!