Question 5
Let $P_1,P_2,\ldots,P_{1993}=P_0$ be distinct points in the $xy$-plane such that (i) both coordinates of each $P_i$ are integers; (ii) for $i=0,1,\ldots,1992$, the segment $P_iP_{i+1}$ contains no lattice point other than its endpoints. Prove that for some $i$ there is a point $Q=(q_x,q_y)$ on $P_iP_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.
Schematic of an odd closed chain with lattice-point vertices; the problem has $1993$ vertices.
Schematic of an odd closed chain with lattice-point vertices; the problem has $1993$ vertices.