Question 106
Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers such that
$$
a_{1}+a_{2}+\cdots+a_{n}=a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2} .
$$
Prove that
$$
\sum_{1 \leq i \lt j \leq n} a_{i} a_{j}\left(1-a_{i} a_{j}\right) \geq 0 .
$$
$$
a_{1}+a_{2}+\cdots+a_{n}=a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2} .
$$
Prove that
$$
\sum_{1 \leq i \lt j \leq n} a_{i} a_{j}\left(1-a_{i} a_{j}\right) \geq 0 .
$$