Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$ Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds:
$$f(x+y)=f(x^{2}+y^{2}).$$

For any positives $a,b$ we have:
\begin{aligned} f(a) &=f\left(\frac{a+b}{2}+\frac{a-b}{2}\right) \\
 &=f\left(\big(\frac{a+b}{2}\big)^2+\big(\frac{a-b}{2}\big)^2\right) \\
 &=f\left(\big(\frac{a+b}{2}\big)^2+\big(\frac{b-a}{2}\big)^2\right) \\
 &=f\left(\frac{a+b}{2}+\frac{b-a}{2}\right) \\
 &=f(b)\end{aligned}
