Found problems: 1513
Find all pairs of positive integers \( (a, b) \) such that \( f(x) = x \) is the only function \( f : \mathbb{R} \to \mathbb{R} \) that satisfies
\[
f^a(x)f^b(y) + f^b(x)f^a(y) = 2xy \quad \text{for all } x, y \in \mathbb{R}.
\]
Here, \( f^n(x) \) represents the function obtained by applying \( f \) \( n \) times to \( x \). That is, \( f^1(x) = f(x) \) and \( f^{n+1}(x) = f(f^n(x))\) for all \(n \geq 1\).
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $$f^{f^{f(x)}(y)}(z)=x+y+z+1$$ for all $x,y,z \in \mathbb{N}$.
[i]Proposed by William Wang.[/i]
Find all functions $f: \mathbb R \to \mathbb R$ such that \[ f( xf(x) + f(y) ) = f^2(x) + y \] for all $x,y\in \mathbb R$.
A function $f$ satisfies the following condition for each $n\in N$: $f (1)+ f (2)+...+ f (n) = n^2 f (n)$.
Find $f (1997)$ if $f (1) = 999$.
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]
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]
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 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 $2022^{2022}$.
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 $f: \mathbb{R} \to \mathbb{R}$, 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 $\mathbb{Z}$ 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 $f:\mathbb{Z}\to\mathbb{Z}$ such that
\[f\left(\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\right)+f(x)=f(f(y))+\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\]
holds for all $x,y\in\mathbb{Z}$.
[i]Proposed by usjl[/i]
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$
$(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$
[i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]
Determine all functions $f : R \to R$ such that $f(x) \ge 0$ for all positive reals $x$, $f(0) = 0$ and for all reals $x, y$
$$f(x + y -xy) = f(x) + f(y) - f(xy).$$
Find all functions $f: R\to R$ such that for all real numbers $x, y$ is satisfied that $$f (x + y) = (f (x))^{ 2013} + f (y).$$
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f \colon \mathbb{N} \to \mathbb{Z}$ such that \[\left\lfloor \frac{f(mn)}{n} \right\rfloor=f(m)\] for all positive integers $m,n$.
[i]Merlijn Staps[/i]
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$