TSTST 2011 Problem 3

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 $$\sqrt{(x-m)^{2}+(y-n)^{2}}<c \log \left(x^{2}+y^{2}+2\right) .$$