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0332 inequalities 3rd edition Round 3 p2

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May 9, 2021
inequalitiesalgebra3rd edition

Problem Statement

Let n be a positive integer and let a1,a2,...,an,b1,b2,...,bn,c2,c3,...,c2na_1, a_2, ..., a_n, b_1, b_2, ... , b_n, c_2, c_3, ... , c_{2n} be 4n−14n - 1 positive real numbers such that ci+j2≥aibjc^2_{i+j} \ge a_ib_j , for all 1≤i,j≤n1 \le i, j \le n. Also let m=max⁡2≤i≤2ncim = \max_{2 \le i\le 2n} c_i. Prove that (m+c2+c3+...+c2n2n)2≥(a1+a2+...+ann)(b1+b2+...+bnn)\left(\frac{m + c_2 + c_3 +... + c_{2n}}{2n} \right)^2 \ge \left(\frac{a_1+a_2 + ... +a_n}{n}\right)\left(\frac{ b_1 + b_2 + ...+ b_n}{n}\right)
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