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4x4 non linear system of equations with cubic radicals

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist A1 day2

September 24, 2021
algebrasystem of equations

Problem Statement

Find all solutions of the system
yz4x23+2wx=0\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0 xz4y3+5wy=0\sqrt[3]{\frac{xz^4}{y}}+5wy=0 xyx3+7wz−1/3=0\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0 x12+1254y5+3432z4=16x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16
where x,y,z≥0x, y, z \ge 0 and w∈Rw \in R
I attached the system, in case I have any typos
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