Problem 1124

Let $a_i$ be the number of boy students in the $i$$\text{th}$ row, then the number of girl students in this row is 75 - a_i$. By the given condition, we have$$\sum_{i=1}^{22} \left( C_{a_i}^2 + C_{75-a_i}^2 \right) \le 11 \times C_{75}^2$$. That is,$$\sum_{i=1}^{22}$ (a_i^2 - 75a_i) $\le$ -30,525$, implying$$\sum_{i=1}^{22}$ (2a_i - 75)^2 $\le$ 1,650$. Using Cauchy's Inequality, we have
$$\left[ \sum_{i=1}^{22} (2a_i - 75) \right]^2 \le 22 \sum_{i=1}^{22} (2a_i - 75)^2 \le 36,300.$$
Then $\sum_{i=1}^{22} (2a_i - 75) < 191$, and $\sum_{i=1}^{22} a_i < \frac{191 + 1,650}{2} < 921$. That means the number of boy students does not exceed $928$.