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 2 of 5: Choose the first corner and orient the order

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In plain words

Cutting at the first visit near $B'$ creates two complementary arcs, one before and one after the excursion.

A'\text{ is the first approached vertex; assume }B'\text{ is approached before }D'

Detailed analysis

Let $A'$ be the first vertex approached by $L$ . After that, both adjacent vertices $B'$ and $D'$ must be approached. Relabel the orientation if necessary and suppose $B'$ is approached first. Let $B$ be the first point of $L$ within $1/2$ of $B'$ ; split $L$ at $B$ into the initial part $L_1$ and the remaining part $L_2$ .