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Prove that the roots of a quadratic are integers

Source: INMO 2007 Question 3

November 2, 2009
quadraticsinequalitiesnumber theory unsolvednumber theory

Problem Statement

Let m m and n n be positive integers such that x^2 \minus{} mx \plus{}n \equal{} 0 has real roots α \alpha and β \beta. Prove that α \alpha and β \beta are integers if and only if [m\alpha] \plus{} [m\beta] is the square of an integer. (Here [x] [x] denotes the largest integer not exceeding x x)
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