Question 80
Find all real $x \ge -1$ such that the inequality
$$
\frac{a_1 + x}{2} \cdot \frac{a_2 + x}{2} \cdot \dots \cdot \frac{a_n + x}{2} \le \frac{a_1 a_2 \dots a_n + x}{2}
$$
holds for all $a_1, \dots, a_n \ge 1$, where $n \ge 2$.
$$
\frac{a_1 + x}{2} \cdot \frac{a_2 + x}{2} \cdot \dots \cdot \frac{a_n + x}{2} \le \frac{a_1 a_2 \dots a_n + x}{2}
$$
holds for all $a_1, \dots, a_n \ge 1$, where $n \ge 2$.