MathDB

Suppose

A_1 A_2 \ldots A_n

is an

n

-sided polygon with

n \geq 3

and

\angle A_j \leq 180^{\circ}

for each

j

(in other words, the polygon is convex or has fewer than

n

distinct sides).
For each

i \leq n

, suppose

\alpha_i

is the smallest possible value of

\angle{A_i A_j A_{i+1}}

where

j

is neither

i

nor

i+1

. (Here, we define

A_{n+1} = A_1

.) Prove that

\alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ}

and determine all equality cases.

Problem Statement