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Non-linear System of Equations with 3 Variables

Source: European Mathematical Cup 2012, Junior Division, Problem 3

July 27, 2013

algebrapolynomialVietasystem of equationsalgebra proposed

Problem Statement

Are there positive real numbers

x

y

and

z

such that

x^4 + y^4 + z^4 = 13\text{,}

x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,}

x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?}

Proposed by Matko Ljulj.