5. Variant 1 Petya wrote 9 different positive integers on the board. It turned out that the arithmetic mean of these numbers is 16. What is the greatest value that the largest number on the board can take
Answer: 108.

Solution: The sum of the given numbers is $9 \cdot 16=144$. Since all the numbers are distinct, the sum of the 8 smallest of them is no less than $1+2+\cdots+8=36$. Therefore, the largest number cannot be greater than $144-36=108$. This is possible: $(1+2+\cdots+8+108): 9=16$.

Variant 2 Petya wrote 9 different positive integers on the board. It turned out that the arithmetic mean of these numbers is 17. What is the largest value that the largest number on the board can take?

Answer: 117.

Variant 3 Petya wrote 9 different positive integers on the board. It turned out that the arithmetic mean of these numbers is 18. What is the largest value that the largest number on the board can take?

Answer: 126.

Variant 4 Petya wrote 9 different positive integers on the board. It turned out that the arithmetic mean of these numbers is 19. What is the largest value that the largest number on the board can take?

Answer: 135.

Variant 5 Petya wrote 9 different positive integers on the board. It turned out that the arithmetic mean of these numbers is 20. What is the largest value that the largest number on the board can take?

Answer: 144.