Problem

Source: Kazakhstan National 2021, Problem 5

Tags: Kazakhstan, Diophantine equation, number theory proposed, algebra, number theory



Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$