Task 1. For what values of p is one of the roots of the equation $x^{2}+p x+18=0$ twice the other?
Answer: 9 or -9.

Solution. Let the roots of the equation be $a$ and $2a$. By Vieta's theorem, $a + 2a = -p$, $a \cdot 2a = 18$. Therefore, $a = \pm 3$, $p = -3a$.

Criteria. For losing the second solution - 2 points are deducted. For an answer without justification - 0 points. A solution without using Vieta's theorem is allowed. In case of full justification - a score of 7 points.