AUSTRIAN-POLISH 2005 Problem 3

Let $a_0, a_1, a_2, ... , a_n$ be real numbers, which fulfill the following two conditions: a) $0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n$ . b) For all $0 \leq i < j \leq n$ holds: $a_j - a_i \leq j-i$ .

Prove that $$\left( \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.$$