Problem 6

Let

S

be a square with side length

100

. Let

L

be a simple closed polygonal path inside

S

, composed of segments

A_0A_1,A_1A_2,\ldots,A_{n-1}A_n

with

A_0=A_n

. Suppose that every point

P

on the boundary of

S

is at distance at most

1/2

from some point of

L

. Prove that there are points

X,Y

L

whose distance is at most

1

and for which the length of the part of

L

between

X

and

Y

is at least

198

Step 5 of 5: Force a long subpath

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In plain words

The path must travel almost the full width to reach $B'$ from either side, so the two nearby points enclose a long excursion.

XB\ge99,\quad YB\ge99\quad\Longrightarrow\quad\operatorname{length}(L_{XY})\ge198

Detailed analysis

Since $B$ is within $1/2$ of $B'$ and $X$ is within $1/2$ of $X'$ , while $X'B'\ge99$ (the side has length $100$ and $X'$ lies on the side away from $D'$ ), $XB\ge X'B'-1\ge99$ . The same argument gives $YB\ge99$ . The arc of $L$ from $X$ to $Y$ through $B$ therefore has length at least $XB+BY\ge198$ .