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sum a_k^3 <= (\sum a _k )^2

Source: All-Russian MO 2002 Regional (R4) 10.4 11.3

September 17, 2024
inequalitiesalgebra

Problem Statement

(10.4) A set of numbers a0,a1,...,ana_0, a_1,..., a_n satisfies the conditions: a0=0a_0 = 0, 0≤ak+1−ak≤10 \le a_{k+1}- a_k \le 1 for k=0,1,..,n−1k = 0, 1, .. , n -1. Prove the inequality ∑k=1nak3≤(∑k=1nak)2\sum_{k=1}^n a^3_k \le \left(\sum_{k=1}^n a_k \right)^2
(11.3) A set of numbers a0,a1,...,ana_0, a_1,..., a_n satisfies the conditions: a0=0a_0 = 0, ak+1≥ak+1a_{k+1} \ge a_k + 1 for k=0,1,..,n−1k = 0, 1, .. , n -1. Prove the inequality ∑k=1nak3≥(∑k=1nak)2\sum_{k=1}^n a^3_k \ge \left(\sum_{k=1}^n a_k \right)^2
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