CWMO 2010 Problem 4

Let $a_1,a_2,..,a_n,b_1,b_2,...,b_n$ be non-negative numbers satisfying the following conditions simultaneously:

(1) $\sum_{i=1}^{n} (a_i + b_i) = 1$ ;

(2) $\sum_{i=1}^{n} i(a_i - b_i) = 0$ ;

(3) $\sum_{i=1}^{n} i^2(a_i + b_i) = 10$ .

Prove that $\text{max}\{a_k,b_k\} \le \dfrac{10}{10+k^2}$ for all $1 \le k \le n$ .