Question 81
Given an integer $n\gt 5$, find all real numbers $a$ for which there exist nonnegative real numbers $x_1,\dots,x_n$ satisfying
$$\sum_{k=1}^n kx_k=a,\qquad\sum_{k=1}^n k^3x_k=a^2,\qquad\sum_{k=1}^n k^5x_k=a^3.$$
$$\sum_{k=1}^n kx_k=a,\qquad\sum_{k=1}^n k^3x_k=a^2,\qquad\sum_{k=1}^n k^5x_k=a^3.$$