1993 APMO 第 5 题

Let $P_{1}, P_{2}, \ldots, P_{1993}=P_{0}$ be distinct points in the $x y$-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 $\left(q_{x}, q_{y}\right)$ on the line segment joining $P_{i}$ with $P_{i+1}$ such that both $2 q_{x}$ and $2 q_{y}$ are odd integers.

令 $P_{1}, P_{2}, \ldots, P_{1993}=P_{0}$ 为 $x y$ 平面上的不同点，具有以下属性：

(i) $P_{i}$ 的两个坐标都是整数，$i=1,2, \ldots, 1993$；

(ii) 在连接 $P_{i}$ 和 $P_{i+1}$ 的线段上，除了 $P_{i}$ 和 $P_{i+1}$ 之外，没有坐标均为整数的点，对于 $i=0,1, \ldots, 1992$。

证明对于某些$i, 0 \leq i \leq 1992$，在连接$P_{i}$和$P_{i+1}$的线段上存在坐标$\left(q_{x}, q_{y}\right)$的点$Q$，使得$2 q_{x}$和$2 q_{y}$都是奇整数。