4. For what least natural $\mathrm{k}$ does the quadratic trinomial

$$
y=k x^{2}-p x+q \text { with natural coefficients } p \text{ and } q \text{ have two }
$$

distinct positive roots less than 1?
Solution. First, let's estimate the coefficient $\mathrm{k}_{\text{from below. Let }} 0k^{2} x_{1}\left(1-x_{1}\right) x_{2}\left(1-x_{2}\right) \geq 1 \Rightarrow k>4 \text{. Thus, we have obtained the }$
$$
$$
lower bound. To show the accuracy of the estimate, it is sufficient to construct an example with $k=5: y=5 x^{2}-5 x+1$.

Recommendations for checking: Constructing only an example, without an estimate - 3 points;

Obtaining only the estimate (without an example) - 4 points.

## Recommendations for checking

Constructing only an example, without an estimate - 3 points;

Obtaining only the estimate (without an example) - 4 points.