Found problems: 1513
2015 Baltic Way, 5
Find all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the equation \[|x|f(y)+yf(x)=f(xy)+f(x^2)+f(f(y))\] for all real numbers $x$ and $y$.
2022 German National Olympiad, 6
Consider functions $f$ satisfying the following four conditions:
(1) $f$ is real-valued and defined for all real numbers.
(2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$.
(3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$.
(4) We have $f(2)=4$.
Prove that:
a) There is a function $f$ with $f(3)=9$ satisfying the four conditions.
b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$.
2025 Kyiv City MO Round 1, Problem 4
Find all functions \( f : \mathbb{N} \to \mathbb{N} \) that satisfy the following condition: for any positive integers \( m \) and \( n \) such that \( m > n \) and \( m \) is not divisible by \( n \), if we denote by \( r \) the remainder of the division of \( m \) by \( n \), then the remainder of the division of \( f(m) \) by \( n \) is \( f(r) \).
[i]Proposed by Mykyta Kharin[/i]
2017 Baltic Way, 5
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x^2y)=f(xy)+yf(f(x)+y)$$ for all real numbers $x$ and $y$.
2015 India IMO Training Camp, 2
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.
2010 Contests, 1
Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.
1990 IMO Longlists, 6
Let function $f : \mathbb Z_{\geq 0}^0 \to \mathbb N$ satisfy the following conditions:
(i) $ f(0, 0, 0) = 1;$
(ii) $f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);$
(iii) when applying above relation iteratively, if any of $x', y', z$' is negative, then $f(x', y', z') = 0.$
Prove that if $x, y, z$ are the side lengths of a triangle, then $\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)}$ is not an integer for any integers $k, m > 1.$
PEN K Problems, 34
Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$:
\[ f(3mn+m+n)=4f(m)f(n)+f(m)+f(n). \]
2005 VJIMC, Problem 3
Find all reals $\lambda$ for which there is a nonzero polynomial $P$ with real coefficients such that
$$\frac{P(1)+P(3)+P(5)+\ldots+P(2n-1)}n=\lambda P(n)$$for all positive integers $n$, and find all such polynomials for $\lambda=2$.
1996 Bosnia and Herzegovina Team Selection Test, 4
Solve the functional equation $$f(x+y)+f(x-y)=2f(x)\cos{y}$$ where $x,y \in \mathbb{R}$ and $f : \mathbb{R} \rightarrow \mathbb{R}$
2025 Ukraine National Mathematical Olympiad, 11.3
Find all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) such that for any real numbers \(x\) and \(y\), the following inequality holds:
\[
f\left(x^2+2y f(x)\right) + (f(y))^2 \leq f\left((x+y)^2\right)
\]
[i]Proposed by Anton Trygub[/i]
2023 Macedonian Mathematical Olympiad, Problem 1
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ we have:
$$xf(x+y)+yf(y-x) = f(x^2+y^2)\,.$$
[i]Authored by Nikola Velov[/i]
2023 IFYM, Sozopol, 4
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 $y$.
2016 District Olympiad, 4
[b]a)[/b] Prove that not all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ f(x-1)+f(x+1) =\sqrt 5f(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
[b]b)[/b] Prove that that all functions $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy the equality
$$ g(x-1)+g(x+1)=\sqrt 3g(x) ,\quad\forall x\in\mathbb{R} , $$
are periodic.
2015 Costa Rica - Final Round, 3
Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that
i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$
ii) $f (f(x)) = f(x)$
where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.
2020-IMOC, A2
Find all function $f:\mathbb{R}^+$ $\rightarrow \mathbb{R}^+$ such that: $f(f(x) + y)f(x) = f(xy + 1) \forall x, y \in \mathbb{R}^+$
2016 Greece Team Selection Test, 3
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 $x,y\in\mathbb{Z}$.
2018 NZMOC Camp Selection Problems, 10
Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$ for all $x, y \in R$.
2018 Silk Road, 2
Find all functions $f:\ \mathbb{R}\rightarrow\mathbb{R}$ such that for any real number $x$ the equalities are true:
$f\left(x+1\right)=1+f(x)$ and $f\left(x^4-x^2\right)=f^4(x)-f^2(x).$
[url=http://matol.kz/comments/3373/show]source[/url]
2005 IMO Shortlist, 2
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property:
\[f(x)f(y)\equal{}2f(x\plus{}yf(x))\]
for all positive real numbers $x$ and $y$.
[i]Proposed by Nikolai Nikolov, Bulgaria[/i]
Fractal Edition 1, P3
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the following two conditions:
\[
\left\{
\begin{array}{ll}
\mbox{If } f(0) = 0, \mbox{ then } f(x) \neq 0 \mbox{ for any non-zero } x. \\
\\
f(x + y)f(y + z)f(z + x) = f(x + y + z)f(xy + yz + zx) - f(x)f(y)f(z) \quad \forall x, y, z \in \mathbb{R}.
\end{array}
\right.
\]
2018 Dutch IMO TST, 4
Let $A$ be a set of functions $f : R\to R$.
For all $f_1, f_2 \in A$ there exists a $f_3 \in A$ such that $f_1(f_2(y) - x)+ 2x = f_3(x + y)$ for all $x, y \in R$.
Prove that for all $f \in A$, we have $f(x - f(x))= 0$ for all $x \in R$.
2022 USA TSTST, 8
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]
2019 Abels Math Contest (Norwegian MO) Final, 3b
Find all real functions $f$ defined on the real numbers except zero, satisfying
$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
2010 Germany Team Selection Test, 3
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]