Found problems: 2349
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[
f(2x + y + f(x + y)) + f(xy) = y f(x)
\]
for all real numbers $x$ and $y$.
Let $f$ be a function defined from $((x,y) : x,y$ real, $xy\ne 0)$ to the set of all positive real numbers such that
$ (i) f(xy,z)= f(x,z)\cdot f(y,z)$ for all $x,y \ne 0$
$ (ii) f(x,yz)= f(x,y)\cdot f(x,z)$ for all $x,y \ne 0$
$ (iii) f(x,1-x) = 1 $ for all $x \ne 0,1$
Prove that
$ (a) f(x,x) = f(x,-x) = 1$ for all $x \ne 0$
$(b) f(x,y)\cdot f(y,x) = 1 $ for all $x,y \ne 0$
The condition (ii) was left out in the paper leading to an incomplete problem during contest.
(a) Prove that there is a unique function $f : Q \to Q$ satisfying:
(i) $f(q)= 1 + f\left(\frac{q}{1-2q}\right)$ for $0<q< \frac12$
(ii) $f(q)= 1 + f(q-1)$ for $1<q\le 2$
(iii) $f(q)f\left(\frac{1}{q}\right)=1$ for all $q\in Q^+$
(b) For this function $f$ , find all $r\in Q^+$ such that $f(r) = r$
Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties:
(i) $f(1) = 2$;
(ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.
Find all $c\in \mathbb{R}$ such that there exists a function $f:\mathbb{R}\to \mathbb{R}$ satisfying $$(f(x)+1)(f(y)+1)=f(x+y)+f(xy+c)$$ for all $x,y\in \mathbb{R}$.
[i]Proposed by Kaan Bilge[/i]
Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.
Find all functions such that $ f: \mathbb{R}^\plus{} \rightarrow \mathbb{R}^\plus{}$ and $ f(x\plus{}f(y))\equal{}yf(xy\plus{}1)$ for every $ x,y\in \mathbb{R}^\plus{}$.
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 $f : \mathbb Q \to \mathbb Q$ be a function such that for any $x,y \in \mathbb Q$, the number $f(x+y)-f(x)-f(y)$ is an integer. Decide whether it follows that there exists a constant $c$ such that $f(x) - cx$ is an integer for every rational number $x$.
[i]Proposed by Victor Wang[/i]
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$,
\[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]
Is there an injective function $f : Z^+ \to Q$ satisfying the equation $f(xy) = f(x) + f(y)$ for all positive integers $x$ and $y$?
Find all functions $f$ from the set $\mathbb{R}$ of real numbers into $\mathbb{R}$ which satisfy for all $x, y, z \in \mathbb{R}$ the identity \[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\]
Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that
forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
Given $X=\{0,a,b,c\}$, let $M(X)=\{f|f: X\to X\}$ denote the set of all functions from $X$ into itself. An addition table on $X$ is given us follows:
$+$ $0$ $a$ $b$ $c$
$0$ $0$ $a$ $b$ $c$
$a$ $a$ $0$ $c$ $b$
$b$ $b$ $c$ $0$ $a$
$c$ $c$ $b$ $a$ $0$
a)If $S=\{f\in M(X)|f(x+y+x)=f(x)+f(y)+f(x)\forall x,y\in X\}$, find $|S|$.
b)If $I=\{f\in M(X)|f(x+x)=f(x)+f(x)\forall x\in X\}$, find $|I|$.
Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ satisfying the equality $ f (x ^ 2-y ^ 2) = (x-y) (f (x) + f (y)) $ for any $ x, y \in \mathbb {R} $.
Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$, such that for all real $x,y$ the following equation holds:$$f(x-0.25)+f(y-0.25)=f(x+\lfloor y+0.25 \rfloor - 0.25)$$
Let $\displaystyle M$ be a set composed of $\displaystyle n$ elements and let $\displaystyle \mathcal P (M)$ be its power set. Find all functions $\displaystyle f : \mathcal P (M) \to \{ 0,1,2,\ldots,n \}$ that have the properties
(a) $\displaystyle f(A) \neq 0$, for $\displaystyle A \neq \phi$;
(b) $\displaystyle f \left( A \cup B \right) = f \left( A \cap B \right) + f \left( A \Delta B \right)$, for all $\displaystyle A,B \in \mathcal P (M)$, where $\displaystyle A \Delta B = \left( A \cup B \right) \backslash \left( A \cap B \right)$.
Let $\mathbb{Z}$ be the set of integers. Find all functions $f: \mathbb{Z}\rightarrow \mathbb{Z}$ such that, for any two integers $m, n$, $$f(m^2)+f(mf(n))=f(m+n)f(m).$$
[i]Proposed by Victor Domínguez and Pablo Valeriano[/i]
Let $a,b$ be two constants within the open interval $(0,\frac{1}{2})$. Find all continous functions $f(x)$ such that \[f(f(x))=af(x)+bx\] holds for all real $x$.
This is much harder than the problems we had in the 1st TST...
A function $g$ is such that for all integer $n$:
$$g(n)=\begin{cases}
1\hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\geq 1 & \\
0 \hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\leq 0 &
\end{cases}$$
A function $f$ is such that for all integers $n\geq 0$ and $m\geq 0$:
$$f(0,m)=0 \hspace{0.5cm} \textrm{and}$$
$$f(n+1,m)=\Bigl(1-g(m)+g(m)\cdot g(m-1-f(n,m))\Bigr)\cdot\Bigl(1+f(n,m)\Bigr)$$
Find all the possible functions $f(m,n)$ that satisfies the above for all integers $n\geq0$ and $m\geq 0$
Find all functions from the reals to the reals satisfying
\[f(xf(y) + x) = xy + f(x)\]
Let $f(x)$ be a continuous function defined on the closed interval $0 \leq x \leq 1$. Let $G(f)$ denote the graph of $f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq$$ x \leq 1, y = f(x) \}$. Let $G_a(f)$ denote the graph of the translated function $f(x - a)$ (translated over a distance $a$), defined by $G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}$. Is it possible to find for every $a, \ 0 < a < 1$, a continuous function $f(x)$, defined on $0 \leq x \leq 1$, such that $f(0) = f(1) = 0$ and $G(f)$ and $G_a(f)$ are disjoint point sets ?
Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.