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$AB>AC$ combogeo

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November 21, 2023

combinatorial geometrycombinatorics

Problem Statement

Let

n\ge 2

be a positive integer, and let

P_1P_2\dots P_{2n}

be a polygon with

2n

sides such that no two sides are parallel. Denote

P_{2n+1}=P_1

. For some point

P

and integer

i\in\{1,2,\ldots,2n\}

, we say that

i

is a

P

-good index if

PP_{i}>PP_{i+1}

, and that

i

is a

P

-bad index if

PP_{i}<PP_{i+1}

. Prove that there's a point

P

such that the number of

P

-good indices is the same as the number of

P

-bad indices.

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