Question 156
Let $t_k = a_1^k + a_2^k + \dots + a_n^k$, where $a_1, a_2, \dots, a_n$ are positive real numbers and $k \in \mathbb{N}$. Prove that
$$
\frac{t_5^2 t_1^6}{15} - \frac{t_4^4 t_2^2 t_1^2}{6} + \frac{t_2^3 t_4^5}{10} \ge 0.
$$
$$
\frac{t_5^2 t_1^6}{15} - \frac{t_4^4 t_2^2 t_1^2}{6} + \frac{t_2^3 t_4^5}{10} \ge 0.
$$