Question 68
Given integer $n \geqslant 4$. Find the maximum of
$$ \frac{\sum_{i=1}^{n} a_i (a_i + b_i)}{\sum_{i=1}^{n} b_i (a_i + b_i)} $$ for non-negative real numbers $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ satisfying
$$ a_1 + a_2 + \dots + a_n = b_1 + b_2 + \dots + b_n \gt 0. $$
$$ \frac{\sum_{i=1}^{n} a_i (a_i + b_i)}{\sum_{i=1}^{n} b_i (a_i + b_i)} $$ for non-negative real numbers $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$ satisfying
$$ a_1 + a_2 + \dots + a_n = b_1 + b_2 + \dots + b_n \gt 0. $$