Estonian Math Competitions

Question

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?

Accepted Answer

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.