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Finding Solutions

Source: Shortlist 2016, Number Theory 5

July 19, 2017
number theoryIMO Shortlist

Problem Statement

Let aa be a positive integer which is not a perfect square, and consider the equation k=x2−ax2−y2.k = \frac{x^2-a}{x^2-y^2}. Let AA be the set of positive integers kk for which the equation admits a solution in Z2\mathbb Z^2 with x>ax>\sqrt{a}, and let BB be the set of positive integers for which the equation admits a solution in Z2\mathbb Z^2 with 0≤x<a0\leq x<\sqrt{a}. Show that A=BA=B.
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