TW-TST 1993 Problem 4

In the Cartesian plane, let $C$ be a unit circle with center at origin $O$ . For any point $Q$ in the plane distinct from $O$ , define $Q'$ to be the intersection of the ray $OQ$ and the circle $C$ . Prove that for any $P\in C$ and any $k\in\mathbb{N}$ there exists a lattice point $Q(x,y)$ with $|x|=k$ or $|y|=k$ such that $PQ'<\frac{1}{2k}$ .