Problem 9.3. The farmer said: "I have $N$ rabbits. Long ears are exactly on 13 of them. And exactly 17 of them can jump far."

The traveler rightly noted: "Therefore, among your rabbits, there are at least 3 rabbits that simultaneously have long ears and can jump far."

What is the largest value that the number $N$ can take?
Answer: 27.

Solution. If $N \geqslant 28$, then the traveler is wrong: it is possible that the farmer had only 2 long-eared rabbits that can jump far. In addition to them, there were 11 long-eared rabbits that cannot jump far, and 15 rabbits without long ears that can jump far. The remaining $N-28$ rabbits neither have long ears nor can jump far.

If $N=27$, then the traveler is right. Indeed, from the condition, it follows that exactly 10 rabbits cannot jump far. Since 13 rabbits have long ears, at least $13-10=3$ of them both have long ears and can jump far.