9.1. Petya wrote 10 integers on the board (not necessarily distinct).

Then he calculated the pairwise products (that is, he multiplied each of the written numbers by each other). Among them, there were exactly 15 negative products. How many zeros were written on the board?
Answer: 2.

Solution. Let there be $A$ positive numbers and $B$ negative numbers on the board. Then $A+B \leq 10$ and $A \cdot B=15$. Since a negative product is obtained when we multiply a negative and a positive number. From this, it is easy to understand that the numbers $A$ and $B$ are 3 and 5 (1). Therefore, $A+B=8$ and there are exactly two zeros on the board.

Criteria. Correct answer without justification - 1 point. Correctly found intermediate relationship (1) - 4 points. Correct answer with justification - 7 points.