Question 142
Given integer $n \ge 2$, find the largest number $\lambda(n)$ with the following property: if a sequence of real numbers $a_0, a_1, a_2, \dots, a_n$ satisfies
$$0 = a_0 \le a_1 \le a_2 \le \dots \le a_n,$$
$$a_i \ge \frac{1}{2}(a_{i+1} + a_{i-1}), \quad i = 1, 2, \dots, n-1,$$
then
$$
\left(\sum_{i=1}^{n} ia_{i}\right)^{2} \geqslant \lambda(n) \sum_{i=1}^{n} a_{i}^{2}.
$$
$$0 = a_0 \le a_1 \le a_2 \le \dots \le a_n,$$
$$a_i \ge \frac{1}{2}(a_{i+1} + a_{i-1}), \quad i = 1, 2, \dots, n-1,$$
then
$$
\left(\sum_{i=1}^{n} ia_{i}\right)^{2} \geqslant \lambda(n) \sum_{i=1}^{n} a_{i}^{2}.
$$