Frankin B.R.

There are 100 non-zero numbers written in a circle. Between every two adjacent numbers, their product was written, and the original numbers were erased. The number of positive numbers did not change. What is the minimum number of positive numbers that could have been written initially?

#
Evaluation. Suppose there were no more than 33 positive numbers. Negative numbers in the product can only be formed if one of the factors is positive, and each positive number can participate in no more than two such products. Therefore, there can be no more than 66 negative numbers. But then the total number of numbers is no more than 99. Contradiction.

Example for 34 positive numbers. Arrange the signs as follows:

![](https://cdn.mathpix.com/cropped/2024_05_06_e123f5c8ab3546621c69g-23.jpg?height=315&width=326&top_left_y=1646&top_left_x=869)

(start with +, then 33 groups of +--, counting counterclockwise). Here, there is one pair of adjacent positive numbers and 33 pairs of adjacent negative numbers - they will give 34 positive products; the rest of the products will be negative.

## Answer

34 numbers.