7.5. One hundred non-zero integers are written in a circle such that each number is greater than the product of the two numbers following it in a clockwise direction. What is the maximum number of positive numbers that can be among these 100 written numbers?
Answer: 50.

Solution: Note that two consecutive numbers cannot both be positive (i.e., natural numbers). Suppose the opposite. Then their product is positive, and the number before them (counterclockwise) is also a natural number. Since it is greater than the product of these two 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.

Therefore, in any pair of adjacent numbers, there is at least one negative number. Dividing our 100 numbers into 50 pairs, we get that there are no fewer than 50 negative numbers, and thus no more than 50 positive numbers. If we alternate a positive number 2 and a negative number -2, we will get an example of the arrangement of 50 positive and 50 negative numbers that satisfy the problem's condition.

Comment: Correct answer without justification - 0 points.

Proving the impossibility of placing two positive numbers next to each other - 3 points.

Proving that there are no more than 50 positive numbers - 1 point.

Providing an example with 50 positive numbers - 3 points.