Let
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
be a square with sides length
![100](/media/m/c/c/c/ccc0563efabf7c1a3d81b0dc63f5b627.png)
. Let
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
be a path within
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
which does not meet itself and which is composed of line segments
![A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n](/media/m/1/d/7/1d7ebeca3a75abc6b1377467fc04bb4c.png)
with
![A_0=A_n](/media/m/8/d/e/8dea061c9aa9461dc66f07bacd40f6c7.png)
. Suppose that for every point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
on the boundary of
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
there is a point of
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
at a distance from
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
no greater than
![1\over2](/media/m/1/7/6/1764dd91ed3188bd458568579b823a3b.png)
. Prove that there are two points
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
of
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
such that the distance between
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
is not greater than
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
and the length of the part of
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
which lies between
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
and
![Y](/media/m/3/b/c/3bc24c5af9ce86a9a691643555fc3fd6.png)
is not smaller than
![198](/media/m/d/7/f/d7f517c639dc5148ac2cae522c827728.png)
.
%V0
Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $1\over2$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.