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Ugly equation in x,y,z

Source: 41st Austrian Mathematical Olympiad, regional competition, problem 2)

April 26, 2010
algebra proposedalgebra

Problem Statement

Solve the following in equation in R3\mathbb{R}^3: 4x4−x2(4y4+4z4−1)−2xyz+y8+2y4z4+y2z2+z8=0.4x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.
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