Found problems: 2349
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have:
$$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a function such that $f(x+y+xy)=f(x)+f(y)+f(xy)$ for all $x, y\in\mathbb{R}$. Prove that $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x, y\in\mathbb{R}$.
Determine all functions $f : \mathbb R \to \mathbb R$ such that $$f(f(x)- f(y)) = f(f(x)) - 2x^2f(y) + f\left(y^2\right),$$ for all reals $x, y$.
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any real numbers $a, b, c, d >0$ satisfying $abcd=1$,\[(f(a)+f(b))(f(c)+f(d))=(a+b)(c+d)\] holds true.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 4)[/i]
Find all functions \( f:\mathbb{Z} \to \mathbb{Z} \) such that
\[
f(x + f(y) - 2y) + f(f(y)) = f(x)
\]
for all integers \( x \) and \( y \).
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$
Find all real numbers $c$ for which there exists a function $f\colon\mathbb R\rightarrow \mathbb R$ such that for each $x, y\in\mathbb R$ it's true that
$$f(f(x)+f(y))+cxy=f(x+y).$$
Let $\alpha$ and $\beta$ be real numbers with $\beta \ne 0$. Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[f(\alpha f(x)+f(y))=\beta x+f(y)\]
holds for all real $x$ and $y$.
[i](Walther Janous)[/i]
The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions:
[list]
[*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$
[*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$
[/list]
Prove that $a_n=n$ for all positive integers $n$.
[i]Proposed by José Alejandro Reyes González[/i]
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(x+f(y^2)) + f(xy) = f(x) + yf(x+y)$$
for all $x, y \in \mathbb{R}$.
[i]Proposed by Md. Fuad Al Alam[/i]
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
Let $f(x)$ be a polynomial of finite degree satisfying
$$(x+9)f(x+1)=(x+3)f(x+3)$$
for all real $x$. If $f(0)=1$, find the value of $f(1)$.
Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals.
[i]Proposed by Paisan Nakmahachalasint, Thailand[/i]
Let $\mathbb R^+$ denote the set of positive real numbers. Find all functions $f:\mathbb R^+\to\mathbb R$ and $g:\mathbb R^+\to\mathbb R$ such that for all $x,y\in\mathbb R^+$, $g(x)-g(y)=(x-y)f(xy)$.
[i]Linus Tang[/i]
Find all functions $f:\mathbb{R}\to \mathbb R$ satisfying
\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]
For all $x,y\in\mathbb R$.
Let \(\mathbb R_{>0}\) denote the set of positive real numbers and \(\mathbb R_{\ge0}\) the set of nonnegative real numbers. Find all functions \(f:\mathbb R\times \mathbb R_{>0}\to \mathbb R_{\ge0}\) such that for all real numbers \(a\), \(b\), \(x\), \(y\) with \(x,y>0\), we have \[f(a,x)+f(b,y)=f(a+b,x+y)+f(ay-bx,xy(x+y)).\]
[i]Proposed by Luke Robitaille[/i]
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 $f:[0,1]\to\mathbb R$ be a continuous function such that $f(0)=f(1)=0$. Prove that the set
$$A:=\{h\in[0,1]:f(x+h)=f(x)\text{ for some }x\in[0,1]\}$$is Lebesgue measureable and has Lebesgue measure at least $\frac12$.
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:
$\bullet$ $f(x)<2$ for all $x\in (0,1)$;
$\bullet$ for all real numbers $x,y$ we have:
$$max\{f(x+y),f(x-y)\}=f(x)+f(y)$$
Proposed by Navid Safaei
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$
[i]Proposed by Warut Suksompong, Thailand[/i]
Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function satisfying
\[f(x+2y)=2f(x)f(y)\]
for every $x,y \in \mathbb{R}$. Prove that $f$ is constant.
Find all functions $f:\mathbb R^+\to \mathbb R^+$ such that $$f(x+y)f(f(x))=f(1+yf(x))$$ for all $x,y\in \mathbb R^+.$
[i]Proposed by Ming Hsiao[/i]
Determine all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ so that $\forall x: x\cdot f(\frac x2) - f(\frac2x) = 1$
Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that:
(i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$.
(ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$.
[i]Proposed by Yang Liu[/i]
Find all surjective functions $f: \mathbb R \to \mathbb R$ such that for every $x,y\in \mathbb R,$ we have
\[f(x+f(x)+2f(y))=f(2x)+f(2y).\]