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Estonia — Open Contests

Algebra Difficulty 4.8 AIME Prove it Estonia

Teacher tells Jüri two nonzero integers $a$ and $b$ such that $b$ is divisible by $a$ . Jüri has to find a nonzero integer $c$ such that $c$ is divisible by $b$ and all solutions of the quadratic equation $ax^2 + bx + c = 0$ are integers. Can Jüri always solve the problem?

Solution

By the conditions of the problem there is an integer $q$ such that $b = aq$ . Let $c = -2aq^2$ ; then $c \neq 0$ and $c$ is divisible by $b$ . The quadratic equation $ax^2 + bx + c = 0$ or $ax^2 + aqx - 2aq^2 = 0$ has solutions $q$ and $-2q$ .

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