# 8. Variant 1.

Each of the 10 students came up with 5 natural numbers. It turned out that each number was thought of by at least three students. What is the maximum number of different numbers that could have been thought of?
Answer: 16.

Solution: In total, the students came up with 50 numbers, with each number being counted at least 3 times. We will prove that there could not have been more than 16 different numbers. If at least 17 different numbers were thought of and each by at least three students, then a total of no less than $17 \cdot 3=51$ numbers were thought of. This is a contradiction. An example where 16 numbers were thought of: Suppose the numbers 1 and 2 were thought of by 4 students each, and numbers 3 to 16 were thought of by 3 students each. Then a total of $2 \cdot 4 + 14 \cdot 3 = 8 + 42 = 50$ numbers were thought of, and each by at least 3 students.