2. Given 50 numbers. It is known that among their pairwise products, exactly 500 are negative. Determine the number of zeros among these numbers.
Solution. Let $m, n$ and $p-$ be the number of negative, zero, and positive numbers among the given 50 numbers. Then from the condition of the problem we have: $m+n+p=50$ and $m \cdot p=500$. It follows that $m$ and $p$ are divisors of 500, the sum of which does not exceed 50. Among all divisors of the number 500, only the pair 20 and 25 has this property, i.e. $m+p=45$, hence $n=5$.