China Western Mathematical Olympiad

Let $a_i$ be the number of boy students in the $i$\text{th}$row,thenthenumberofgirlstudentsinthisrowis$75 - a_i$.Bythegivencondition,wehave$\sum_{i=1}^{22} \left( C_{a_i}^2 + C_{75-a_i}^2 \right) \le 11 \times C_{75}^2$.Thatis,$\sum_{i=1}^{22} (a_i^2 - 75a_i) \le -30,525$,implying$\sum_{i=1}^{22} (2a_i - 75)^2 \le 1,650$.UsingCauch{y}^{\prime }sInequality,wehave$\sum_{i=1}^{22} (2a_i - 75) < 191$,and$\sum_{i=1}^{22} a_i < \frac{191 + 1,650}{2} < 921$.Thatmeansthenumberofboystudentsdoesnotexceed$928$.