3

Part of 1983 Spain Mathematical Olympiad

Problems(1)

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

Source: Spanish Mathematical Olympiad 1983 P3

A semicircle of radius

r

is divided into

n + 1

equal parts and any point

k

of the division with the ends of the semicircle forms a triangle

A_k

. Calculate the limit, as

n

tends to infinity, of the arithmetic mean of the areas of the triangles.

geometrycombinatorial geometrycombinatoricsarea of a triangle