Found problems: 66
2014 Middle European Mathematical Olympiad, 1
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\]
holds for all $x,y \in \mathbb{R}$.
Russian TST 2018, P1
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.
2025 Vietnam Team Selection Test, 1
Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$ holds for all positive rational numbers $x, y$.
1994 French Mathematical Olympiad, Problem 5
Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$,
$$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$.
(b) Calculate $f(n)$ for any natural number $n$.
2017-IMOC, A4
Show that for all non-constant functions $f:\mathbb R\to\mathbb R$, there are two real numbers $x,y$ such that
$$f(x+f(y))>xf(y)+x.$$
2000 Brazil Team Selection Test, Problem 2
Find all functions $f:\mathbb R\to\mathbb R$ such that
(i) $f(0)=1$;
(ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$;
(iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.
2018-IMOC, A1
Find all functions $f:\mathbb Q\to\mathbb Q$ such that for all $x,y,z,w\in\mathbb Q$,
$$f(f(xyzw)+x+y)+f(z)+f(w)=f(f(xyzw)+z+w)+f(x)+f(y).$$
2008 VJIMC, Problem 2
Find all functions $f:(0,\infty)\to(0,\infty)$ such that
$$f(f(f(x)))+4f(f(x))+f(x)=6x.$$
2019 Romanian Masters In Mathematics, 5
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 $y$.
2004 VJIMC, Problem 2
Find all functions $f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}$ such that
$1$. $f(x,0)=f(0,x)=x$ for all $x\in\mathbb R_{\ge0}$,
$2$. $f(f(x,y),z)=f(x,f(y,z))$ for all $x,y,z\in\mathbb R_{\ge0}$ and
$3$. there exists a real $k$ such that $f(x+y,x+z)=kx+f(y,z)$ for all $x,y,z\in\mathbb R_{\ge0}$.
1993 French Mathematical Olympiad, Problem 3
Let $f$ be a function from $\mathbb Z$ to $\mathbb R$ which is bounded from above and satisfies $f(n)\le\frac12(f(n-1)+f(n+1))$ for all $n$. Show that $f$ is constant.
2017-IMOC, A2
Find all functions $f:\mathbb N\to\mathbb N$ such that
\begin{align*}
x+f(y)&\mid f(y+f(x))\\
f(x)-2017&\mid x-2017\end{align*}
2017-IMOC, N1
If $f:\mathbb N\to\mathbb R$ is a function such that
$$\prod_{d\mid n}f(d)=2^n$$holds for all $n\in\mathbb N$, show that $f$ sends $\mathbb N$ to $\mathbb N$.
2022 Indonesia TST, A
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
\[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.
2000 Mongolian Mathematical Olympiad, Problem 4
Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions:
(i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$;
(ii) $f(f(f(0)))=0$.
Prove that $f(0)=0$.
1999 Mongolian Mathematical Olympiad, Problem 1
Suppose that a function $f:\mathbb R\to\mathbb R$ is such that for any real $h$ there exist at most $19990509$ different values of $x$ for which $f(x)\ne f(x+h)$. Prove that there is a set of at most $9995256$ real numbers such that $f$ is constant outside of this set.