JBMO 2023 | Problem 2 - 25 June 2023

Problem 2.
Prove that for all nonnegative real numbers x,y and z, which are not all equal to 0, the following inequality holds:

$\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geqslant 3.$

Determine all triples (x,y,z) for which the equality holds.