SILK-ROAD 2019 Problem 2

Let $a_1,$ $a_2,$ $\ldots,$ $a_ {99}$ be positive real numbers such that $ia_j + ja_i \ge i + j$ for all $1 \le i <j \le 99.$ Prove , that $(a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!$ .