Question 135
Let $(a_n)_{n=1}^{\infty}$ be a strictly increasing sequence of positive real numbers such that the inequality
$$
a_n(a_n - 2a_{n-1}) + a_{n-1}(a_{n-1} - 2a_{n-2}) \ge 0
$$
holds for all $n \ge 3$. Prove that for all $n \ge 2$ the inequality
$$
a_n \ge a_{n-1} + a_{n-2} + \dots + a_1
$$
holds as well.
$$
a_n(a_n - 2a_{n-1}) + a_{n-1}(a_{n-1} - 2a_{n-2}) \ge 0
$$
holds for all $n \ge 3$. Prove that for all $n \ge 2$ the inequality
$$
a_n \ge a_{n-1} + a_{n-2} + \dots + a_1
$$
holds as well.