MathDB

M2802

Part of Kvant 2024

Problems(1)

Arithmetic means on a circle

Source: Caucasus MO 2024, Juniors P7

3/15/2024
The positive numbers a1,a2,…,a2024a_1, a_2, \ldots , a_{2024} are placed on a circle clockwise in this order. Let AiA_i be the arithmetic mean of the number aia_i and one or several following it clockwise. Prove that the largest of the numbers A1,A2,…,A2024A_1, A_2, \ldots , A_{2024} is not less than the arithmetic mean of all numbers a1,a2,…,a2024a_1, a_2, \ldots , a_{2024}.
combinatoricsalgebra