Hello. I have an example for you. I'm curious how. Yesterday I was on the mathematical competition. One example I can not solve. I want to know how. Can you help me, please? Consider a convex polygon of 1415 sides, which circumference is 2001 cm. Prove that between its peaks, there are 3 such that vertices form a triangle with an area less than 1 cm2. So... We know it is 1415-gon. The sum of its interior angles is π(n-2) rad (where n is 1415). We can calculate the average length of a side: 2001/1415 cm... And the last what I think: there is Pigeonhole principle but i don't know how to do it. Thanks very much for your ideas and sorry for my bad English.