Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.5 AIME, harder Prove it Estonia

In a math period, the teacher asks pupils to solve quadratic equations of the form $x^2 + px + q = 0$ where $p$ and $q$ are some integers. The teacher obtains every new equation by either increasing by 1 or decreasing by 1 the value of either $p$ or $q$ in the equation just solved. In the initial equation, $p = 2020$ and $q = 2010$ , whereas in the last equation, $p = 2010$ and $q = 2020$ . Is it definitely true that both solutions of at least one equation solved during the period are integers?

Solution

In the first equation, one has $p - q = 10$ , while in the last equation, one has $p - q = -10$ . At each step, either $p$ or $q$ changes exactly by 1, whence also $p - q$ changes exactly by 1. Thus at some step one must have an equation where $p - q = 1$ , or equivalently, $p = q + 1$ . The equation $x^2 + (q + 1)x + q = 0$ has integral solutions $-1$ and $-q$ .

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