Question 30
Given the positive numbers $a_1, a_2, \dots, a_n$, such that $n \gt 2$ and $a_1 + a_2 + \dots + a_n = 1$, prove that the inequality
$$
\frac{a_2 a_3 \dots a_n}{a_1 + n - 2} + \frac{a_1 a_3 \dots a_n}{a_2 + n - 2} + \frac{a_1 a_2 a_4 \dots a_n}{a_3 + n - 2} + \dots + \frac{a_1 a_2 \dots a_{n-1}}{a_n + n - 2} \le \frac{1}{(n-1)^2}.
$$
Does holds.
$$
\frac{a_2 a_3 \dots a_n}{a_1 + n - 2} + \frac{a_1 a_3 \dots a_n}{a_2 + n - 2} + \frac{a_1 a_2 a_4 \dots a_n}{a_3 + n - 2} + \dots + \frac{a_1 a_2 \dots a_{n-1}}{a_n + n - 2} \le \frac{1}{(n-1)^2}.
$$
Does holds.