Determine all functions satisfying $xf(y) + yf(x) = (x+y)f(x^2+y^2)$
1071
Determine all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $xf(y)+yf(x)=(x+y)f(x^2+y^2)$ for all $x,y\in\mathbb{N}$ (contest question)
75448
Find all function satisfying $f(f(n))+f(n)=2n+3k$
27809
Find the function equation $f(x+f(y)+yf(x))=y+f(x)+xf(y)$
155363
Find all integers $(x, y)$ such that $1 + 2^x + 2^{2x + 1} = y^2$
66772
$f(f(x)f(y))+f(x+y)=f(xy)$
29647
Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?
111467
find all natural solution that satisfy $x^2+y^2 = 3z^2$
83894
$f(x^2) = xf(x)$ implies that $ f(x) = mx$?
122329
Determine all functions $f(x)$ such that $f(f(x+y))=f(x)+f(y)$
136475
d'Alembert-like functional equation: $f(x+y)+g(x-y)=\lambda f(x)g(y)$
69734
how to find all functions such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$
50111
non-continuous function satisfies $f(x+y)=f(x)+f(y)$
132093
Solve $f(f(x))=f(x)$
42699
Solve for $f(x)$ if $f(f(x))=6x-f(x)$
100900
Finding maxima and minima of $f(x,y)= x^4 + y^4 - 2x^2 - 2y^2 + 4xy$
69707
If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
76458
Solve the following functional equation $f(xf(y))+f(yf(x))=2xy$