Problem 10.5. Greedy Vovochka has 25 classmates. For his birthday, he brought 200 candies to the class. Vovochka's mother, to prevent him from eating them all himself, told him to distribute the candies so that any 16 of his classmates would collectively have at least 100 candies. What is the maximum number of candies Vovochka can keep for himself while fulfilling his mother's request?
Answer: 37.

Solution. Among all 25 classmates, select 16 people with the smallest number of candies given. Note that among them, there is a person who received no less than 7 candies (otherwise, if they all received no more than 6 candies, then in total they received no more than $16 \cdot 6=96$ candies, which is less than 100). Then the remaining $25-16=9$ classmates received no fewer than 7 candies. Thus, in total, at least $100+9 \cdot 7=163$ candies were given out. Accordingly, Vovochka left himself no more than $200-163=37$ candies.

Note also that Vovochka could have left himself exactly 37 candies if he gave the other 163 to his classmates as follows: 13 classmates received 7 candies each, and 12 classmates received 6 candies each. Then any 16 people together have at least $6 \cdot 12+7 \cdot 4=100$ candies, and Vovochka indeed gave out a total of $13 \cdot 7+12 \cdot 6=163$ candies.