KöMaL 2022 C1726

Prove that if $x$, $y$, $z$ are real numbers such that $\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=1, \text{then} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=0.$ Find all real numbers satisfying this condition.