Question 38
A set $a_1 \ge a_2 \ge \dots \ge a_{100n}$ of positive numbers has the following property. For any subset of $2n + 1$ numbers, the sum of $n$ maximal numbers in this subset is greater than the sum of the remaining $n + 1$ numbers. Prove that
$$
(n + 1)(a_1 + a_2 + \dots + a_n) \gt a_{n+1} + a_{n+2} + \dots + a_{100n}.
$$
$$
(n + 1)(a_1 + a_2 + \dots + a_n) \gt a_{n+1} + a_{n+2} + \dots + a_{100n}.
$$