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Ukrainian Mathematical Olympiad, Final Round

Algebra Difficulty 5.2 AIME, harder Prove it Ukraine

Find all pairs of real numbers $x$ and $y$ such that
$\frac{x-2}{y} + \frac{5}{xy} = \frac{4-y}{x} - \frac{|y-2x|}{xy}.$

Solution

Зауважимо, що при допустимих значеннях $x$ і $y$ вихідна рівність рівносильна рівності
$(x-1)^2 + (y-2)^2 + |y-2x| = 0.$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.