Find all such monotonous function $f:[0,+\infty)\to \mathbb R$, such that for any real $r\ge0$ and $\varphi \in \left[\frac{\pi}6, \frac{\pi}4\right]$, $$f(r \cos \varphi)+f(r \sin \varphi)=f(r)\text.$$
My work so far:
If $\varphi \in \left[\frac{\pi}6, \frac{\pi}4\right]$ then $$\frac{\sqrt2}2\le\cos \varphi\le\frac{\sqrt3}2$$ $$\frac12\le\sin \varphi\le\frac{\sqrt2}2$$
If $r=0$ then $f(0)=0$
If $\varphi= \frac{\pi}{4}$ then $$f(r)=4f\left(\frac r2\right)$$
$f(x)=cx^2$ for $x\ge0$ and $c\in\mathbb R$