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arithmetic mean of areas of triangles by n+1 points on semicircle and 2 ends

Source: Spanish Mathematical Olympiad 1983 P3

December 23, 2022
geometrycombinatorial geometrycombinatoricsarea of a triangle

Problem Statement

A semicircle of radius rr is divided into n+1n + 1 equal parts and any point kk of the division with the ends of the semicircle forms a triangle AkA_k. Calculate the limit, as nn tends to infinity, of the arithmetic mean of the areas of the triangles.