5. In a school there are 300 boys and 300 girls, divided into 5 classes, each with the same number of students. It is known that there are at least 33 boys and 33 girls in each class. A boy and a girl from the same class may form a group to enter a contest, and each student may only belong to one group. What is the maximum number of groups that can be guaranteed to form?
(1 mark)
某校有男生和女生各 300 名，他們被分成 5 班，每班人數相同。已知每班均最少有男生和女生各 33 名。同班的一名男生和一名女生可組隊參加一項比賽, 而每名學生只可隸屬一隊。保證能夠組成的隊伍數目的最大值是多少？
(1 分)
5. 192
5. Let $b_{i}$ and $g_{i}$ (where $1 \leq i \leq 5$ ) be the number of boys and girls in the $i$-th class respectively. Since there are $600 \div 5=120$ students in each class and at least 33 boys and girls, each $b_{i}$ and $g_{i}$ is between 33 and 87 inclusive. Consider the matrix:
$$
\left(\begin{array}{lllll}
b_{1} & b_{2} & b_{3} & b_{4} & b_{5} \\
g_{1} & g_{2} & g_{3} & g_{4} & g_{5}
\end{array}\right)
$$

The sum of each row is 300 , while the sum of each column is 120 . If we circle the smaller number in each column, the sum of the circled numbers is the number of teams which can be formed.

By the pigeonhole principle, at least three numbers are circled in the same row. Without loss of generality assume $b_{1}, b_{2}$ and $b_{3}$ are circled. Note that $b_{1}+b_{2}+b_{3} \geq 300-2 \times 87=126$. On the other hand, each circled number in the fourth and fifth column is at least 33 . The sum of the circled numbers is thus at least $126+33+33=192$. Equality is possible as the matrix
$$
\left(\begin{array}{lllll}
42 & 42 & 42 & 87 & 87 \\
78 & 78 & 78 & 33 & 33
\end{array}\right)
$$
provides one such example. The answer is thus 192.