9.5. Three hundred non-zero integers are written in a circle such that each number is greater than the product of the next three numbers in the clockwise direction. What is the maximum number of positive numbers that can be among these 300 written numbers?
Answer: 200.

Solution: Note that three consecutive numbers cannot all be positive (i.e., natural numbers). Suppose the opposite. Then their product is positive, and the number preceding them (counterclockwise) is also a natural number. Since it is greater than the product of these three natural numbers, it is greater than each of them. Continuing this reasoning, we will conclude that all numbers are natural, and if we move counterclockwise, the numbers will increase. However, when the "circle closes," we will arrive at a contradiction.

Thus, in any triplet of adjacent numbers, there is at least one negative number. Dividing our 300 numbers into 100 triplets, we get that there are no fewer than 100 negative numbers, and therefore, no more than 200 positive numbers. If we alternate two positive numbers 2 and one negative number -2, we will get an example of the arrangement of 200 positive and 100 negative numbers that satisfy the problem's condition.

Comment: A correct answer without justification - 0 points.

Proving the impossibility of placing three positive numbers consecutively - 3 points. Proving that there are no more than 200 positive numbers - another 1 point.

Providing an example with 200 positive numbers - 3 points.