# Task 1. (10 points)

How many natural numbers $n$ not exceeding 2017 exist such that the quadratic trinomial $x^{2}+x-n$ can be factored into linear factors with integer coefficients

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# Solution.

According to the problem, \(x^{2}+x-n=(x-a)(x-b)\). Therefore, \(ab = -n\), which means the numbers \(a\) and \(b\) have different signs and are not zero. Without loss of generality, we assume that \(a \geq 0\). Since \(a + b = -1 \Rightarrow b = -1 - a\), we have

\[
ab = -n = a(-1 - a) \Rightarrow a^{2} + a = n \leq 2017 \Rightarrow a \leq 44
\]

Thus, we obtain 44 pairs of numbers \(a\) and \(b\) that satisfy the given conditions.

Answer. 44.

| Content of the criterion | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | 10 |
| All logical steps of the solution are provided, including the estimate for one of the roots and the relationship between the roots of the quadratic trinomial. The answer is incorrect. | \(\pm\) | 7 |
| The estimate for one of the roots of the quadratic trinomial is correct. The answer is incorrect or missing. | \(+/ 2\) | 5 |
| The relationship between \(n\) and the roots of the quadratic trinomial is shown. Incorrect estimates for one of the roots and/or the number of numbers \(n\) satisfying the problem's condition are provided. | \(\mp\) | 2 |
| The solution does not meet any of the criteria described above. | \(-/ 0\) | 0 |
| Maximum score |  | 10 |