1982 IMO 第 6 题

A6 (VIE 1) ${ }^{\text {IMO6 }}$ Let $S$ be a square with sides of length 100 and let $L$ be a path within $S$ that does not meet itself and that is composed of linear segments $A_{0} A_{1}, A_{1} A_{2}, \ldots, A_{n-1} A_{n}$ with $A_{0} \neq A_{n}$. Suppose that for every point $P$ of the boundary of $S$ there is a point of $L$ at a distance from $P$ not greater than $\frac{1}{2}$. Prove that there are two points $X$ and $Y$ in $L$ such that the distance between $X$ and $Y$ is not greater than 1 and the length of that part of $L$ that lies between $X$ and $Y$ is not smaller than 198.

A6 (VIE 1) ${ }^{\text {IMO6 }}$ 设 $S$ 为边长为 100 的正方形，设 $L$ 为 $S$ 内不与自身相交且由线性线段 $A_{0} A_{1}、A_{1} A_{2}、\ldots、A_{n-1} A_{n}$ 和 $A_{0} \neq 组成的路径A_{n}$。假设对于 $S$ 边界上的每个点 $P$，都有一个 $L$ 点，其与 $P$ 的距离不大于 $\frac{1}{2}$。证明$L$中有两个点$X$和$Y$，使得$X$和$Y$之间的距离不大于1，并且$L$中位于$X$和$Y$之间的部分的长度不小于198。