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Romania NMO 2023 Grade 9 P1

Source: Romania National Olympiad 2023

April 14, 2023
algebraquadratic equation

Problem Statement

We consider the equation x2+(a+b−1)x+ab−a−b=0x^2 + (a + b - 1)x + ab - a - b = 0, where aa and bb are positive integers with a≤ba \leq b.
a) Show that the equation has 22 distinct real solutions.
b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and b<2a.b < 2a.
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