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ASU 267 All Soviet Union MO 1978, Σ(a_n - b_n)^2

Source:

July 11, 2019
algebraSuminequalitiesAverage

Problem Statement

Given a1,a2,...,ana_1, a_2, ... , a_n. Define bk=a1+a2+...+akkb_k = \frac{a_1 + a_2 + ... + a_k}{k} for 1≤k≤n.1 \le k\le n. Let C=(a1−b1)2+(a2−b2)2+...+(an−bn)2,D=(a1−bn)2+(a2−bn)2+...+(an−bn)2C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + ... + (a_n - b_n)^2, D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + ... + (a_n - b_n)^2 Prove that C≤D≤2CC \le D \le 2C.
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