MathDB

A convex

n

-gon

A_1A_2\dots A_n

(n>3)

is divided into triangles by non-intersecting diagonals. For every vertex the number of sides issuing from it is even, except for the vertices

A_{i_1},A_{i_2},\dots,A_{i_k}

, where

1\leq i_1<\dots<i_k\leq n

. Prove that

k

is even and

n\equiv i_1-i_2+\dots+i_{k-1}-i_k\pmod3

k>0

and n\equiv0\pmod3\mbox{ for }k=0. Note that this leads to generalization of one recent Tournament of towns problem about triangulating of square.

Problem Statement