Question 6
A6 (JAP). Let $n$ be an integer, $n\ge3$. Let $x_1,x_2,\ldots,x_n$ be real numbers such that $x_i\lt x_{i+1}$ for $1\le i\le n-1$. Prove that
$$\frac{n(n-1)}2\sum_{i\lt j}x_ix_j\gt \left(\sum_{i=1}^{n-1}(n-i)x_i\right)\left(\sum_{j=2}^n(j-1)x_j\right).$$
$$\frac{n(n-1)}2\sum_{i\lt j}x_ix_j\gt \left(\sum_{i=1}^{n-1}(n-i)x_i\right)\left(\sum_{j=2}^n(j-1)x_j\right).$$