Question 106
Let $a_1, a_2, a_3, \dots, b_1, b_2, b_3, \dots$ and $c_1, c_2, c_3, \dots$ be three sequences containing the numbers $+1$ and $-1$ only. Prove that the following inequality holds:
$$
3 \sum_{i=1}^{2014} a_i b_i + 3 \sum_{i=1}^{2014} b_i c_i \le \sum_{i=1}^{2014} a_i^2 + \sum_{i=1}^{2014} b_i^2 + \sum_{i=1}^{2014} c_i^2 + 3 \sum_{i=1}^{2014} a_i c_i.
$$
$$
3 \sum_{i=1}^{2014} a_i b_i + 3 \sum_{i=1}^{2014} b_i c_i \le \sum_{i=1}^{2014} a_i^2 + \sum_{i=1}^{2014} b_i^2 + \sum_{i=1}^{2014} c_i^2 + 3 \sum_{i=1}^{2014} a_i c_i.
$$