Question 130
a) Determine all functions $f: \mathbb{R} \longrightarrow \mathbb{Z}$ such that
$$
f(f(y)-x)^{2}+f(x)^{2}+f(y)^{2}=f(y)(1+2 f(f(y)))
$$
for all real numbers $x$ and $y$.
b) Determine all functions $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that
$$
f(f(y)-x)^{2}+f(x)^{2}+f(y)^{2}=f(y)(1+2 f(f(y)))
$$
for all real numbers $x$ and $y$.
$$
f(f(y)-x)^{2}+f(x)^{2}+f(y)^{2}=f(y)(1+2 f(f(y)))
$$
for all real numbers $x$ and $y$.
b) Determine all functions $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that
$$
f(f(y)-x)^{2}+f(x)^{2}+f(y)^{2}=f(y)(1+2 f(f(y)))
$$
for all real numbers $x$ and $y$.