Maths Olympiad Prep

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Algebra Difficulty 5.2 AIME, harder Prove it Ukraine

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

Solution

Зауважимо, що при допустимих значеннях xx і yy вихідна рівність рівносильна рівності
(x−1)2+(y−2)2+∣y−2x∣=0. (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.