Found problems: 313
Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds:
$$f(2^x+2y) =2^yf(f(x))f(y).$$
Find all functions $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions:
1) $|f(x)|\ge 1$,
2) $f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real values of $x $ and $y$.
Find all functions $f: R \to R$, such that equality $f (xf (y) - yf (x)) = f (xy) - xy$ holds for all $x, y \in R$.
Let $f : Z_+ \to Z_+$ such that:
a) $f(n + 1) > f(n)$ for all $n \in Z_+$
b) $f(n + f(m)) = f(n) + m + 1$ for all $n,m \in Z_+$
Find $f(2006)$.
A function $f$ satisfies $f(x)+x f(1-x) = x^2$ for all real $x$. Determine $f$ .
Find all functions $f$ from the real numbers to the real numbers such that $f(xy) \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$.
Find all functions $f: R\to R$, which satisfy the equation $f (xy + f (x)) = xf (y) + f (x)$.
Let $ a, b, c, d \in \mathbb {R}, $ $ b \neq0. $
Find the functions of the $ f: \mathbb{R} \to \mathbb{R} $ such that $ f (x + d \cdot f (y)) = ax + by + c, $ for all $ x, y \in \mathbb{R}. $
Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.
Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$
$$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.
Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.
Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?
Let $A = \{1,2,...,m + n\}$, where $m,n$ are positive integers, and let the function f : $A \to A$ be defined by:
$f(m) = 1$, $f(m+n) = m+1$ and $f(i) = i+1$ for all the other $i$.
(a) Prove that if $m$ and $n$ are odd, then there exists a function $g : A \to A$ such that $g(g(a)) = f(a)$ for all $a \in A$.
(b) Prove that if $m$ is even, then there is a function $g : A\to A$ such that $g(g(a))=f(a)$ for all $a \in A$ is and only if $n = m$.
A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.
Find all functions $f : R \to R$ that satisfy $f(x + y^2 - f(y)) = f(x)$ for all $x,y \in R$.
(Vo Quoc Ba Can)
Find all functions $f : Z^+ \to Z^+$ such that $f(k + 1) >f(f(k))$ for $k > 1$, where $Z^+$ is the set of positive integers.
Find all functions $f : R \to R$ such that $f( 2x^3 + f (y)) = y + 2x^2 f (x)$ for all real numbers $x, y$.
Determine all functions $f : R \to R$ such that
$$f(x)f(y) = 2f(x + y) + 9xy \ \ \forall x, y \in R.$$
Is there a function $f(x)$ defined for all $x$ and such that for some $a$ and all $x$ holds the equality
$$f(x) + f(2x^2 - 1) = 2x + a?$$
Find the function $\displaystyle{f : \mathbb{R}-\left\{ 0\,\right\} \rightarrow \mathbb{R} }$ such that $$f(x)+f(1-\frac{1}{x}) = \frac{1}{x},\,\,\, \forall x \in \mathbb{R}- \{ 0, 1\,\}$$
[i](-InnoXenT-)[/i]
The real polynomial $f \in R[X]$ has an odd degree and it is given that $f$ is co-prime with $g(x) = x^2 - x - 1$ and
$$f(x^2 - 1) = f(x)f(-x), \forall x \in R.$$
Prove that $f$ has at least two complex non-real roots.
Find all functions $f : R \to R$ such that $$xf(x+xy)= xf(x)+ f(x^2)f(y)$$ for all $x,y \in R$.
Find all functions $f : R \to R$ such that for all $x, y \in R$:
$$f(x + yf(x + y)) = y^2 + f(xf(y + 1)).$$
Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$.