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 4 of 5: Make the two points close

In plain words

Both points lie in the same half-unit neighborhood of $X'$ , so they are at most one unit apart.

XY\le XX'+X'Y\le\frac12+\frac12=1

Detailed analysis

Choose $Y$ on $L_2$ within $1/2$ of $X'$ . Since $X$ is within $1/2$ of $X'$ by definition, the triangle inequality gives $XY\le1$ .