Problem - 3827
Let $f$ be a function such that $$ \sqrt {x - \sqrt { x + f(x) } } = f(x) , $$for $x > 1$. In that domain, $f(x)$ has the form $\frac{a+\sqrt{cx+d}}{b},$ where $a,b,c,d$ are integers and $a,b$ are relatively prime. Find $a+b+c+d.$
For convenience, let $y=f(x)$, then
\begin{align*}
\sqrt{x- \sqrt{x + y}} &= y\\
x- \sqrt{x + y}&= y^{2}\\
\sqrt{x + y}&= x- y^{2}\\
x + y&= x^{2} -2x y^{2}+y^{4}\\
y^{4}-2x y^{2} -y + (x^{2}-x)&=0
\end{align*}
Among four solutions, only $y=f(x)=\frac{\sqrt{4x-3}-1}{2}$ works.