Question 163
Let $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ be non-negative numbers satisfying the following conditions simultaneously:
$$
(1) \quad \sum_{i=1}^{n} (a_i + b_i) = 1;
$$
$$
(2) \quad \sum_{i=1}^{n} i(a_i - b_i) = 0;
$$
$$
(3) \quad \sum_{i=1}^{n} i^2 (a_i + b_i) = 10.
$$
Prove that $\max\{a_k, b_k\} \le \frac{10}{10+k^2}$ for all $1 \le k \le n$.
$$
(1) \quad \sum_{i=1}^{n} (a_i + b_i) = 1;
$$
$$
(2) \quad \sum_{i=1}^{n} i(a_i - b_i) = 0;
$$
$$
(3) \quad \sum_{i=1}^{n} i^2 (a_i + b_i) = 10.
$$
Prove that $\max\{a_k, b_k\} \le \frac{10}{10+k^2}$ for all $1 \le k \le n$.