PL-OM 2017

Integers $a_1, a_2, \ldots, a_n$ satisfy $$1<a_1<a_2<\ldots < a_n < 2a_1.$$ If $m$ is the number of distinct prime factors of $a_1a_2\cdots a_n$ , then prove that $$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$

Three sequences $(a_0, a_1, \ldots, a_n)$ , $(b_0, b_1, \ldots, b_{n})$ , $(c_0, c_1, \ldots, c_{2n})$ of non-negative real numbers are given such that for all $0\leq i,j\leq n$ we have $a_ib_j\leq (c_{i+j})^2$ . Prove that $$\sum_{i=0}^n a_i\cdot\sum_{j=0}^n b_j\leq \left( \sum_{k=0}^{2n} c_k\right)^2.$$