Found problems: 1513
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$.
[i]Proposed by B.J. Venkatachala, India[/i]
Find all continuous functions $f : \mathbb{R} \to \mathbb{R}$ such that$$f(-x)+\int \limits_0^xtf(x-t)dt=x,\ \forall\ x\in \mathbb{R}$$
Find all continuous functions $f : (0,+\infty) \to (0,+\infty)$ such that if $a, b, c$ are the lengths of the sides of any triangle then it is satisfied that $$\frac{f(a+b-c)+f(b+c-a)+f(c+a-b)}{3}=f\left(\sqrt{\frac{ab+bc+ca}{3}}\right)$$
Let $G$ be a finite group and $f: G \to G$ a map, such that $f (xy) = f (x) f (y)$ for all $x, y \in G$ and $f (x) = x^{-1}$ for more than $\frac34$ of all $x \in G$ is fulfilled. Show that $f (x) =x^{-1}$ even holds for all $x \in G$ holds.
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that, for all real numbers $x$ and $y$, $$f(x)(x+f(f(y))) = f(x^2)+xf(y)$$
Does there exist a function $f : N\to N$ such that $f ( f (n-1)) = f (n+1)- f (n)$ for all $n \ge 2$?
Find all functions $g : R \to R$, for which there exists a strictly increasing function $f : R \to R$ such that
$f(x + y) = f(x)g(y) + f(y)$.
Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.
Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that for all $m\in\mathbb{Z}$: [list][*] $f(m+8) \le f(m)+8$, [*] $f(m+11) \ge f(m)+11$.[/list]
a) Prove that for no real a such that $0 \le a <1$ there exists a function defined on the set of all positive numbers and taking values in the same set, satisfying for all positive $x$ the equality $$f\left(f(x)+\frac{1}{f(x)}\right)=x+a \,\,\,\,\,\,\, (*) $$
b) Prove that for any $a>1$ there are infinitely many functions defined on the set of all positive numbers, with values in the same set, satisfying ($*$) for all positive x.
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following equation for all $x, y \in \mathbb{R}$.
$$(x+y)f(x-y) = f(x^2-y^2).$$
Let $R$ be the set of real numbers and $f : R \to R$ be a function that satisfies:
$$f(xy) + y + f(x + f(y)) = (y + 1)f(x),$$ for all real numbers $x, y$.
a) Determine the value of $f(0)$.
b) Prove that $f(x) = 2-x$ for every real number $x$.
Let $S = \{2, 3, 4, \dots\}$ be the set of positive integers greater than 1. Find all functions $f : S \to S$ that satisfy
\[
\text{gcd}(a, f(b)) \cdot \text{lcm}(f(a), b) = f(ab)
\]
for all pairs of integers $a, b \in S$.
Clarification: $\text{gcd}(a,b)$ is the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ is the least common multiple of $a$ and $b$.
Find all functions $f: R \to R$ such that $f (f (x) f (y)) = xy$ and there is no $k \in R -\{0,1,-1\}$ such that $f (k) = k$.
Find all real polynomials $p(x) $ such that $p(x-1)p(x+1)= p(x^2-1)$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have
\[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]
Suppose a non-identically zero function $f$ satisfies $f\left(x\right)f\left(y\right)=f\left(\sqrt{x^2+y^2}\right)$ for all $x$ and $y$. Compute $$f\left(1\right)-f\left(0\right)-f\left(-1\right).$$
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that the following equation holds for every real $x,y$:
\[ f(f(x) + y) = \lfloor x + f(f(y)) \rfloor. \]
[b]Note:[/b] $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.
Find all functions $f: (0, \infty) \to (0, \infty)$ such that
\begin{align*}
f(y(f(x))^3 + x) = x^3f(y) + f(x)
\end{align*}
for all $x, y>0$.
[i]Proposed by Jason Prodromidis, Greece[/i]
Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$
Find all functions $f : R \to R$ which satisfy $f \left(\frac{\sqrt3}{3} x\right) = \sqrt3 f(x) - \frac{2\sqrt3}{3} x$
and $f(x)f(y) = f(xy) + f \left(\frac{x}{y} \right) $ for all $x, y \in R$, with $y \ne 0$
Let $P$ be a given quadratic polynomial. Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $$f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}.$$
Find all functions $f:R -\{0,2\} \to R$ that satisfy for all $x \ne 0,2$ $$f(x) \cdot \left(f\left(\sqrt[3]{\frac{2+x}{2-x}}\right) \right)^2=\frac{x^3}{4}$$
Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ satisfying
\[f\bigl(x + y^2 f(y)\bigr) = f\bigl(1 + yf(x)\bigr)f(x)\]
for any positive reals $x$, $y$, where $\mathbb{R}^+$ is the collection of all positive real numbers.
[i]Proposed by Ming Hsiao.[/i]
Let f be a function that satisfies the following conditions:
$(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$.
$(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions;
$(iii)$ $f(0) = 1$.
$(iv)$ $f(1987) \leq 1988$.
$(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$.
Find $f(1987)$.
[i]Proposed by Australia.[/i]