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Inequality with a(j)-a(i)<=j-i

Source: Austrian-Polish 2005, Problem 3

July 5, 2015
inequalitiesn-variable inequality

Problem Statement

Let a0,a1,a2,...,ana_0, a_1, a_2, ... , a_n be real numbers, which fulfill the following two conditions: a) 0=a0≤a1≤a2≤...≤an0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n. b) For all 0≤i<j≤n0 \leq i < j \leq n holds: aj−ai≤j−ia_j - a_i \leq j-i.
Prove that (∑i=0nai)2≥∑i=0nai3.\left( \displaystyle \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.
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