Found problems: 1513
For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by
\[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\]
Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
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]
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying
\[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\]
for all real numbers $x$ and $y$.
Let $f: \mathbb {R} \to \mathbb {R}$ be a concave function and $g: \mathbb {R} \to \mathbb {R}$ be a continuous function . If $$ f (x + y) + f (x-y) -2f (x) = g (x) y^2 $$for all $x, y \in \mathbb {R}, $ prove that $f $ is a second degree polynomial.
Let $\mathbb Z$ be the set of integers. We consider functions $f :\mathbb Z\to\mathbb Z$ satisfying
\[f\left(f(x+y)+y\right)=f\left(f(x)+y\right)\]
for all integers $x$ and $y$. For such a function, we say that an integer $v$ is [i]f-rare[/i] if the set
\[X_v=\{x\in\mathbb Z:f(x)=v\}\]
is finite and nonempty.
(a) Prove that there exists such a function $f$ for which there is an $f$-rare integer.
(b) Prove that no such function $f$ can have more than one $f$-rare integer.
[i]Netherlands[/i]
The function $f(x,y)$, defined on the set of all non-negative integers, satisfies
(i) $f(0,y)=y+1$
(ii) $f(x+1,0)=f(x,1)$
(iii) $f(x+1,y+1)=f(x,f(x+1,y))$
Find f(3,2005), f(4,2005)
Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties:
(a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$.
(b) We have $f\left(2\right) = 0$.
(c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$.
[b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[ f(xy+f(x)) + f(y) = xf(y) + f(x+y) \]
for all real numbers $x$ and $y$.
Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]
Find all functions $f$ $:$ $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ :
$$(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)$$
Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.)
[i]Proposed by Sean Li[/i]
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy
\[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\]
for every $x, y \in \mathbb{R}$, and $f(kx) > kf(x)$ for every $x \in \mathbb{R}$ and $k \in \mathbb{R}$, such that $k > 1$.
$f,g:\mathbb{R}\rightarrow\mathbb{R}$ find all $f,g$ satisfying $\forall x,y\in \mathbb{R}$:
\[g(f(x)-y)=f(g(y))+x.\]
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$ for all $x,y\in\mathbb{R}^+$. Show that $f$ is [i]unbounded[/i], i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.
Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition:
\[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]
Find all polynomials $f(x)$ with real coefficients for which
\[f(x)f(2x^2) = f(2x^3 + x).\]
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real numbers $a, b, c$, we have
\[
f(a+b+c)f(ab+bc+ca) - f(a)f(b)f(c) = f(a+b)f(b+c)f(c+a).
\]
[i]Proposed by Mainak Ghosh and Rijul Saini[/i]
Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that
$$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\in \mathbb{R}$.
[i]Proposed by Adrian Beker[/i]
Find all functions$ f : R_+ \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$
(Folklore)
[hide=PS]Using search terms [color=#f00]+ ''f(x+f(y))'' + ''f(x)+y[/color]''
I found the same problem [url=https://artofproblemsolving.com/community/c6h1122140p5167983]in Q[/url], [url=https://artofproblemsolving.com/community/c6h1597644p9926878]continuous in R[/url], [url=https://artofproblemsolving.com/community/c6h1065586p4628238]strictly monotone in R[/url] , [url=https://artofproblemsolving.com/community/c6h583742p3451211 ]without extra conditions in R[/url]
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Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(x^3\plus{}y^3)\equal{}xf(x^2)\plus{}yf(y^2)\] for all real numbers $ x$ and $ y$.
[i]Hery Susanto, Malang[/i]
Let $ f(0) \equal{} f(1) \equal{} 0$ and
\[ f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2}, \quad n \equal{} 0, 1, 2, \ldots\]
Show that the numbers $ f(1989), f(1990), f(1991)$ are divisible by $ 13.$
Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying
\[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\]
for all integers $a$ and $b$