Something something Newton's Sums

Assume there does exist positive real number solutions to the equations. Let $S_p = \displaystyle \sum_{i = 0}^n x_i^p,$ where $p$ is a positive integer. By the Cauchy-Schwarz Inequality,

$\begin{aligned} S_{d}S_{d + 2} &= (x_1^d + x_2^d + \dots + x_n^d)(x_1^{d + 2} + x_2^{d + 2} + \dots + x_n^{d + 2}) \\ &\geq \left ( \sqrt{x_1^{2d + 2}} + \sqrt{x_2^{2d + 2}} + \dots + \sqrt{x_n^{2d +2}} \right )^2 \\ &= (x_1^{d + 1} + x_2^{d + 1} + \dots + x_n^{d + 1})^2 \\ &= S_{d + 1}^2. \end{aligned}$

Thus, $S_7S_9 \geq S_8^2,$ and $S_8S_{10} \geq S_9^2.$ Multiplying these together gives $S_7S_{10} \geq S_8S_9.$ We know that $S_7 = 7, S_8 = 8, S_9 = 9, S_{10} = 10.$ Plugging these in gives $7(10) \geq 8(9),$ which is clearly false. Thus, there are no positive real numbers $x_1, x_2, \dots, x_n$ such that the system of equations is true.

Note that we can extend the argument to prove that, for all positive integers $a, b$ such that $b \geq a + 2,$ we have $S_aS_b \geq S_{a + 1}S_{b - 1}.$