Question 1
Let $P_{1}, P_{2}, \ldots, P_{1993}=P_{0}$ be distinct points in the $xy$-plane with the following properties:
(i) both coordinates of $P_{i}$ are integers, for $i=1,2, \ldots, 1993$;
(ii) there is no point other than $P_{i}$ and $P_{i+1}$ on the line segment joining $P_{i}$ with $P_{i+1}$ whose coordinates are both integers, for $i=0,1, \ldots, 1992$.
Prove that for some $i, 0 \leq i \leq 1992$, there exists a point $Q$ with coordinates ($q_{x}, q_{y}$) on the line segment joining $P_{i}$ with $P_{i+1}$ such that both $2q_{x}$ and $2q_{y}$ are odd integers.
(i) both coordinates of $P_{i}$ are integers, for $i=1,2, \ldots, 1993$;
(ii) there is no point other than $P_{i}$ and $P_{i+1}$ on the line segment joining $P_{i}$ with $P_{i+1}$ whose coordinates are both integers, for $i=0,1, \ldots, 1992$.
Prove that for some $i, 0 \leq i \leq 1992$, there exists a point $Q$ with coordinates ($q_{x}, q_{y}$) on the line segment joining $P_{i}$ with $P_{i+1}$ such that both $2q_{x}$ and $2q_{y}$ are odd integers.