Found problems: 1513
2019 China Team Selection Test, 4
Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that
1) $f(0,x)$ is non-decreasing ;
2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ;
3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .
2011 ELMO Problems, 4
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
2007 Stars of Mathematics, 1
Prove that there exists just one function $ f:\mathbb{N}^2\longrightarrow\mathbb{N} $ which simultaneously satisfies:
$ \text{(1)}\quad f(m,n)=f(n,m),\quad\forall m,n\in\mathbb{N} $
$ \text{(2)}\quad f(n,n)=n,\quad\forall n\in\mathbb{N} $
$ \text{(3)}\quad n>m\implies (n-m)f(m,n)=nf(m,n-m), \quad\forall m,n\in\mathbb{N} $
2012 Albania Team Selection Test, 5
Let $f:\mathbb R^+ \to \mathbb R^+$ be a function such that: \[
x,y > 0 \qquad f(x+f(y)) = yf(xy+1).
\]
a) Show that $(y-1)*(f(y)-1) \le 0$ for $y>0$.
b) Find all such functions that require the given condition.
2005 India IMO Training Camp, 2
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2023 Korea Summer Program Practice Test, P2
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x)^2 + |y|) = x^2 + f(y)$$
2024-IMOC, A4
find all function $f:\mathbb{R} \to \mathbb{R}$ such that
\[f(x^3-xf(y)^2)=xf(x+y)f(x-y)\]
holds for all real number $x$, $y$.
[i]Proposed by chengbilly[/i]
2024 Ukraine National Mathematical Olympiad, Problem 4
Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following:
$$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$
[i]Proposed by Mykhailo Shtandenko[/i]
2019 Estonia Team Selection Test, 3
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.
2023 IMO, 3
For each integer $k\geq 2$, determine all infinite sequences of positive integers $a_1$, $a_2$, $\ldots$ for which there exists a polynomial $P$ of the form \[ P(x)=x^k+c_{k-1}x^{k-1}+\dots + c_1 x+c_0, \] where $c_0$, $c_1$, \dots, $c_{k-1}$ are non-negative integers, such that \[ P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k} \] for every integer $n\geq 1$.
2012 Putnam, 3
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that
(i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$
(ii) $ f(0)=1,$ and
(iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite.
Prove that $f$ is unique, and express $f(x)$ in closed form.
2017 Taiwan TST Round 1, 3
Find all injective functions $ f:\mathbb{N} \to \mathbb{N} $ such that $$ f^{f\left(a\right)}\left(b\right)f^{f\left(b\right)}\left(a\right)=\left(f\left(a+b\right)\right)^2 $$ holds for all $ a,b \in \mathbb{N} $. Note that $ f^{k}\left(n\right) $ means $ \underbrace{f(f(\ldots f}_{k}(n) \ldots )) $
2008 Indonesia TST, 2
Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.
2002 Austrian-Polish Competition, 7
Find all real functions $f$ definited on positive integers and satisying:
(a) $f(x+22)=f(x)$,
(b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$
for all positive integers $x$ and $y$.
2024 ELMO Shortlist, A3
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$
[i]Andrew Carratu[/i]
2014 District Olympiad, 4
Find all functions $f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast}$ with
the properties:
[list=a]
[*]$ f(m+n) -1 \mid f(m)+f(n),\quad \forall m,n\in\mathbb{N}^{\ast} $
[*]$ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast} $[/list]
2012 Balkan MO Shortlist, N3
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
2023 Indonesia TST, A
Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied
\[f(x+y) + f(x)f(y) = f(xy) + 1 \]
$\forall x, y \in \mathbb{R}$
2014 BMT Spring, 8
Suppose an integer-valued function $f$ satisfies
$$\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1}$$
for all non-negative integers $n$. Determine $\sum_{n=0}^\infty\frac{f(n)}{2^n}$.
2008 Mathcenter Contest, 2
Find all the functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the functional equation $$f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y)$$ for every $x,y\in\mathbb{R}$ and $f(2008) =f(-2008)$
[i](nooonuii)[/i]
2014 France Team Selection Test, 4
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 $n$.
2016 Turkey Team Selection Test, 5
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .
2016 Taiwan TST Round 1, 4
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 $x,y\in\mathbb{Z}$.
2016 Benelux, 3
Find all functions $f :\Bbb{ R}\to \Bbb{Z}$ such that $$\left( f(f(y) - x) \right)^2+ f(x)^2 + f(y)^2 = f(y) \cdot \left( 1 + 2f(f(y)) \right),$$ for all $x, y \in \Bbb{R}.$
2023 Federal Competition For Advanced Students, P2, 1
Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.