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Beautiful combinatorial geometry (Iran TST)

Source: Iranian TST 2019, second exam day 2, problem 5

April 25, 2019

combinatorial geometrycombinatoricsperimeter

Problem Statement

Let

P

be a simple polygon completely in

C

, a circle with radius

1

, such that

P

does not pass through the center of

C

. The perimeter of

P

36

. Prove that there is a radius of

C

that intersects

P

at least

6

times, or there is a circle which is concentric with

C

and have at least

6

common points with

P

.
Proposed by Seyed Reza Hosseini