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x+y^2 +z^3 = 3, y+z^2 +x^3 = 3, z+x^2 +y^3 = 3

Source: 1986 Swedish Mathematical Competition p4

March 28, 2021
system of equationsSystemalgebra

Problem Statement

Prove that x=y=z=1x = y = z = 1 is the only positive solution of the system {x+y2+z3=3y+z2+x3=3z+x2+y3=3\left\{ \begin{array}{l} x+y^2 +z^3 = 3\\ y+z^2 +x^3 = 3\\ z+x^2 +y^3 = 3\\ \end{array} \right.
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