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 1 of 5: Define approach to the boundary

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In plain words

The path comes within a half-unit of the entire boundary, so its order of visits to the corners carries topological information.

\text{Call }P'\text{ approached if some }P\in L\text{ satisfies }PP'\le\frac12

Detailed analysis

A boundary point $P'$ is approached when $L$ contains a point within $1/2$ of it. The hypothesis says every boundary point, hence all four vertices and every boundary subsegment, is approached.