7.3. Anya wrote down 100 numbers in her notebook. Then Sonya wrote down in her notebook all the pairwise products of the numbers written by Anya. Artem noticed that there were exactly 2000 negative numbers in Sonya's notebook. How many zeros did Anya initially write down in her notebook?
Answer: 10 zeros.

Solution: Let Anya write down $n$ positive numbers, $m$ negative numbers, and $100-n-m$ zeros in her notebook. Then, by the condition, $n m=2000$, since a negative number can only be obtained by multiplying numbers of different signs.

Let's list all the divisors of the number $2000=2^{4} * 5^{3}$:

|  | $2^{0}$ | $2^{1}$ | $2^{2}$ | $2^{3}$ | $2^{4}$ |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $5^{0}$ | 1 | 2 | 4 | 8 | 16 |
| $5^{1}$ | 5 | 10 | 20 | 40 | 80 |
| $5^{2}$ | 25 | 50 | 100 | 200 | 400 |
| $5^{3}$ | 125 | 250 | 500 | 1000 | 2000 |

They can be paired such that the product of each pair is 2000. Let's write them in pairs and list their sums:

| 1 | 2 | 4 | 5 | 8 | 10 | 16 | 20 | 25 | 40 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 2000 | 1000 | 500 | 400 | 250 | 200 | 125 | 100 | 80 | 50 |


| 2001 | 1002 | 504 | 405 | 258 | 210 | 141 | 120 | 105 | 90 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |

Only in one of these cases is the sum less than 100, so this is the case that is realized. Thus, there are 90 non-zero numbers and 10 zeros.

Criteria: Only the answer - 0 points.

Only the answer with verification - 1 point.

Not all cases of divisors are considered - no more than 3 points.

No explanation of why there are no other divisors - deduct 1 point.