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A regular 1981-gon has 64 vertices. Prove that there exists a trapezium with vertices at the marked points.

On a plane, n randomly placed lines are given. What is the number of triangles formed by them?

A rabbit, preparing for the arrival of guests, hung lightbulbs in three corners of his polygonal hole. Winnie the Pooh and Piglet came and noticed that lights did not illuminate all the pots of honey which were in the rabbit hole. When they reached for some honey, two of the lightbulbs broke. The rabbit moved the remaining light bulb into some corner so that the whole hole was lit up. Is this possible? $($If yes, draw an example, if not, justify the answer.$)$