Question 12
Let $a_1, a_2, \dots, a_n$ be non-negative integers such that for all real numbers $x_1 \gt x_2 \gt x_3 \gt \dots \gt x_n \gt 0$ with $x_1 + x_2 + \dots + x_n \lt 1$ it holds that $\sum_{k=1}^n a_k x_k^3 \lt 1$.
Show that
$$
na_1 + (n-1)a_2 + \dots + (n-j+1)a_j + \dots + a_n \le \frac{n^2(n+1)^2}{4}.
$$
G. Baron, Vienna
Show that
$$
na_1 + (n-1)a_2 + \dots + (n-j+1)a_j + \dots + a_n \le \frac{n^2(n+1)^2}{4}.
$$
G. Baron, Vienna