Olympiad problems

2022 European Mathematical Cup, 3

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$ for all real numbers $x$, $y$ and $z$ with $x+y+z=0$.

1993 All-Russian Olympiad, 3

Tags: domain , functional equation , function , algebra

Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$.

2018 CMI B.Sc. Entrance Exam, 3

Tags: functional equation , combinatorics , number theory

Let $f$ be a function on non-negative integers defined as follows $$f(2n)=f(f(n))~~~\text{and}~~~f(2n+1)=f(2n)+1$$ [b](a)[/b] If $f(0)=0$ , find $f(n)$ for every $n$. [b](b)[/b] Show that $f(0)$ cannot equal $1$. [b](c)[/b] For what non-negative integers $k$ (if any) can $f(0)$ equal $2^k$ ?

2019 Belarus Team Selection Test, 8.2

Tags: functional equation , function , algebra

Let $\mathbb Z$ be the set of all integers. Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying the following conditions: 1. $f(f(x))=xf(x)-x^2+2$ for all $x\in\mathbb Z$; 2. $f$ takes all integer values. [i](I. Voronovich)[/i]

2025 Macedonian Balkan MO TST, 3

Tags: algebra , functional equation

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy \[f(xf(y) + f(x)) = f(x)f(y) + 2f(x) + f(y) - 1,\] for every $x, y \in \mathbb{R}$, and $f(kx) > kf(x)$ for every $x \in \mathbb{R}$ and $k \in \mathbb{R}$, such that $k > 1$.

1948 Putnam, B4

Tags: integration , functional equation

For what $\lambda$ does the equation $$ \int_{0}^{1} \min(x,y) f(y)\; dy =\lambda f(x)$$ have continuous solutions which do not vanish identically in $(0,1)?$ What are these solutions?

1998 Romania Team Selection Test, 1

Tags: algebra , functional equation , function

Find all monotonic functions $u:\mathbb{R}\rightarrow\mathbb{R}$ which have the property that there exists a strictly monotonic function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x+y)=f(x)u(x)+f(y) \] for all $x,y\in\mathbb{R}$. [i]Vasile Pop[/i]

2009 IMAC Arhimede, 4

Tags: functional , functional equation , algebra

Let $m,n \in Z, m\ne n, m \ne 0, n \ne 0$ . Find all $f: Z \to R$ such that $f(mx+ny)=mf(x)+nf(y)$ for all $x,y \in Z$ .

2021 USA IMO Team Selection Test, 3

Tags: algebra , functional equation

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the inequality \[ f(y) - \left(\frac{z-y}{z-x} f(x) + \frac{y-x}{z-x}f(z)\right) \leq f\left(\frac{x+z}{2}\right) - \frac{f(x)+f(z)}{2} \] for all real numbers $x < y < z$. [i]Proposed by Gabriel Carroll[/i]

2011 QEDMO 10th, 1

Tags: functional , algebra , functional equation

Find all functions $f: R\to R$ with the property that $xf (y) + yf (x) = (x + y) f (xy)$ for all $x, y \in R$.

2016 Germany Team Selection Test, 2

Tags: functional equation

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 $x,y\in\mathbb{Z}$.

1998 Italy TST, 1

Tags: algebra , functional equation , function , functional

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

2015 Bulgaria National Olympiad, 4

Tags: functional equation , algebra

Find all functions $f:\mathbb{R^+}\to\mathbb {R^+} $ such that for all $x,y\in R^+$ the followings hold: $i) $ $f (x+y)\ge f (x)+y $ $ii) $ $f (f (x))\le x $

2016 HMIC, 3

Tags: algebra , hmmt , functional inequality , functional equation

Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$. [i]Pakawut Jiradilok[/i]

1990 Polish MO Finals, 1

Tags: algebra , functional equation , function

Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ that satisfy \[ (x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2) \]

1986 IMO, 2

Tags: function , functional equation , algebra

Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$.

2004 India IMO Training Camp, 3

Tags: function , algebra , trigonometry , system of equations , functional equation

Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

2021 Thailand Online MO, P8

Tags: greatest common divisor , functional equation , number theory

Let $\mathbb N$ be the set of positive integers. Determine all functions $f:\mathbb N\times\mathbb N\to\mathbb N$ that satisfy both of the following conditions: [list] [*]$f(\gcd (a,b),c) = \gcd (a,f(c,b))$ for all $a,b,c \in \mathbb{N}$. [*]$f(a,a) \geq a$ for all $a \in \mathbb{N}$. [/list]