Ukrainian Mathematical Olympiad

Question

Positive numbers x and y satisfy equation x^2 + y^2 + 8xyx+y = 16. Prove that x + y = 4.

Accepted Answer

(x^2 + y^2)(x + y) + 8xy = 16(x + y), ( (x + y)^2 - 2xy )(x + y) - 16(x + y) + 8xy = 0, (x + y)^3 - 16(x + y) - 2xy(x + y) + 8xy = 0, (x + y)((x + y)^2 - 16) - 2xy(x + y - 4) = 0, (x + y)(x + y - 4)(x + y + 4) - 2xy(x + y - 4) = 0, (x + y - 4)((x + y)(x + y + 4) - 2xy) = 0. For x > 0 and y > 0 (x + y)(x + y + 4) - 2xy = x^2 + y^2 + 4(x + y) > 0.