CWMO 2013 Problem 2

Let the integer $n \ge 2$ , and the real numbers $x_1,x_2,\cdots,x_n\in \left[0,1\right]$ .Prove that$$\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.$$