Saudi Arabian Mathematical Competitions, 2015

The original inequality is equivalent to
$$2 \sum_{1 \leq i<j \leq n} a_{i} a_{j} \geq 2 \sum_{1 \leq i<j \leq n} a_{i}^{2} a_{j}^{2},$$
or
$$\left(\sum_{i=1}^{n} a_{i}\right)^{2}-\sum_{i=1}^{n} a_{i}^{2} \geq\left(\sum_{i=1}^{n} a_{i}^{2}\right)^{2}-\sum_{i=1}^{n} a_{i}^{4} .$$
Using the given hypothesis, it can be rewritten as
$$a_{1}^{4}+a_{2}^{4}+\cdots+a_{n}^{4} \geq a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2} .$$
To prove this last inequality, we have from Holder's inequality
$$\left(a_{1}^{4}+a_{2}^{4}+\cdots+a_{n}^{4}\right)\left(a_{1}+a_{2}+\cdots+a_{n}\right)^{2} \geq\left(a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\right)^{3} .$$
Using the given hypothesis again, we get the result.