ÖMO 2013 Problem 4

For a positive integer $n$ , let $a_1, a_2, \ldots a_n$ be nonnegative real numbers such that for all real numbers $x_1>x_2>\ldots>x_n>0$ with $x_1+x_2+\ldots+x_n<1$ , the inequality $\sum_{k=1}^na_kx_k^3<1$ holds. Show that $$na_1+(n-1)a_2+\ldots+(n-j+1)a_j+\ldots+a_n\leqslant\frac{n^2(n+1)^2}{4}.$$