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 3 of 5: Select the last boundary point approached before

B

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In plain words

The boundary point where the first arc stops being able to approach is approached from the other arc.

X'\in A'D'\text{ is the point closest to }D'\text{ approached by }L_1

Detailed analysis

On the side $A'D'$ , let $X'$ be the point closest to $D'$ approached by $L_1$ , and let $X$ be the corresponding point of $L_1$ . Since $D'$ is approached by $L$ but not by $L_1$ , the segment from $X'$ to $D'$ is nonempty and every point of its relative interior must be approached by $L_2$ ; by compactness, $X'$ itself is approached by $L_2$ .