Found problems: 1513
Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.
Determine all functions $f : R \to R$ such that for all $x, y \in R$ holds
$$f(xf(x) + f(y)) = y + f(x)^2$$
The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$.
Find all functions $f$ from the reals to the reals such that
\[f\left(f(x)+y\right)=2x+f\left(f(y)-x\right)\]
for all real $x,y$.
Let $ S$ be the set of all real numbers strictly greater than −1. Find all functions $ f: S \to S$ satisfying the two conditions:
(a) $ f(x \plus{} f(y) \plus{} xf(y)) \equal{} y \plus{} f(x) \plus{} yf(x)$ for all $ x, y$ in $ S$;
(b) $ \frac {f(x)}{x}$ is strictly increasing on each of the two intervals $ \minus{} 1 < x < 0$ and $ 0 < x$.
Let $\mathbb{R}_{\neq 0}$ be the set of nonzero real numbers. Find all functions $f:\mathbb{R}_{\neq 0}\rightarrow\mathbb{R}_{\neq 0}$ such that, for all $x,y\in\mathbb{R}_{\neq 0}$, \[(x-y)f(y^2)+f\left(xy\,f\left(\frac{x^2}{y}\right)\right)=f(y^2f(y)).\]
Let f be a function from the set Z of all integers into itself, that satisfies the condition for all $m \in Z$,
$$f(f(m)) =-m. \ \ (1)$$
Then:
(a) $f$ is a mutually unique mapping, i.e. a simple mapping of the set $Z$ onto the set $Z$ ,
(b) for all $m \in Z$ holds that $f(-m) = -f(m)$ ,
(c) $f(m) = 0$ if and only if $m = 0$ .
Prove these statements and construct an example of a mapping f that satisfies condition (1).
Find all function $f:\mathbb{R}\to \mathbb{R}$ such that
\[f(x)f(y)=f(xy-1)+yf(x)+xf(y)\]
for all $x,y \in \mathbb{R}$
Find all integer-valued polynomials $$f, g : \mathbb{N} \rightarrow \mathbb{N} \text{ such that} \; \forall \; x \in \mathbb{N}, \tau (f(x)) = g(x)$$
holds for all positive integer $x$, where $\tau (x)$ is the number of positive factors of $x$
[i]Proposed By - ckliao914[/i]
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for every $x,y \in \mathbb{R}$
$$f(x^{4}-y^{4})+4f(xy)^{2}=f(x^{4}+y^{4})$$
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 $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$
\[f(x+f(xy))+y=f(x)f(y)+1\]
[i]Ukraine[/i]
For some positive integer $n,$ Elmo writes down the equation
\[x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.\]
Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation
\[f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.\]
Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$. Is it possible that there is no integer $k$ (possibly depending on $f$) such that $f^k(x)=x$ for all $x$?
[i]Srinivas Arun[/i]
Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$
i) $f(ax) = a^2f(x)$ and
ii) $f(f(x)) = a f(x).$
Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$
\[f(x+f(xy))+y=f(x)f(y)+1\]
[i]Ukraine[/i]
Find all functions $f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}$ such that
\[ f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}. \]
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
Find all functions $f : Z \to Z$ such that $x f (2f (y) - x) + y^2 f (2x - f (y)) = \frac{(f (x))^2}{x} + f (y f (y))$ ,
for all $x, y \in Z$, $x \ne 0$.
Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying
\[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\
1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\
+ f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\
+ f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases}
\]
for all nonnegative integers $ p$, $ q$, $ r$.
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+f(f(n))+f(n)=3n.\]
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(xf(x + y)) = yf(x) + 1$$
holds for all $x, y \in \mathbb{R}^{+}$.
[i]Proposed by Nikola Velov, North Macedonia[/i]
Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$f(x)f\left(yf(x)-1\right)=x^2f(y)-f(x),$$for all $x,y \in \mathbb{R}$.
a) Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that\[f(x-f(y))=f(f(x))-f(y)-1\]holds for all $x,y\in\mathbb{Z}$. (It is [url=https://artofproblemsolving.com/community/c6h1268817p6621849]2015 IMO Shortlist A2 [/url])
b) The same question for if \[f(x-f(y))=f(f(x))-f(y)-2\] for all integers $x,y$
Let $\mathcal M_n(\mathbb R)$ denote the set of all real $n\times n$ matrices. Find all surjective functions $f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}$ which satisfy
$$f(XY)\le\min\{f(X),f(Y)\}$$for all $X,Y\in\mathcal M_n(\mathbb R)$.
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$