Found problems: 1513
Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that
$$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$
holds for all real numbers $x; y$
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(0)+1=f(1)$ and for any real numbers $x$ and $y$,
$$ f(xy-x)+f(x+f(y))=yf(x)+3 $$
Let $f: Z ^+ \to R$, such that $f (1) = 2018$ and $f (1) + f (2) + ...+ f (n) = n^2f (n)$, for all $n> 1$. Find the value $f (2017)$.
Let $f$ be a function of the positive reals in the positive reals, such that
$$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$
(a) Find $f(1)$.
(b) Find $f(x)$.
Let $\mathbb{P}$ be the set of positive rational numbers and let $f:\mathbb{P}\to\mathbb{P}$ be such that $$f(x)+f\left(\frac{1}{x}\right)=1$$ and $$f(2x)=2f(f(x))$$ for all $x\in\mathbb{P}$.
Find, with proof, an explicit expression for $f(x)$ for all $x\in \mathbb{P}$.
Let $X = Q-\{-1,0,1\}$. We consider the function $f: X\to X$ given by $f (x) = x -\frac{1}{x} .$ Is there an $a \in X$ such that for every natural number n there is a $y \in X$ with $f (f (...( f (y)) ...)) = a$ where $f$ occurs exactly $n$ times on the left side?
Determine all functions $f : R \times R \to R$ satisfying the equation $f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010$ for all real numbers $x, y$ and for all nonzero $t$
The function $f: N \to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \in\{0,1\}$ for all $m,n \in N$. Determine $f (2014)$. $N_0=\{0,1,2,...\}$
Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations
\[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]
Let $f:Z \to N -\{0\}$ such that:
$f(x + y)f(x-y) = (f(x)f(y))^2$ and $f(1)\ne 1$.
Provethat $\log_{f(1)}f(z)$ is a perfect square for every integer $z$.
Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.
Let $n$ be a positive integer. Find all functions $f\colon \mathbb{R}\rightarrow \mathbb{R}$ that satisfy the equation
$$
(f(x))^n f(x+y) = (f(x))^{n+1} + x^n f(y)
$$
for all $x ,y \in \mathbb{R}$.
For what $\lambda$ does the equation
$$ \int_{0}^{1} \min(x,y) f(y)\; dy =\lambda f(x)$$
have continuous solutions which do not vanish identically in $(0,1)?$ What are these solutions?