Olympiad problems

2000 Swedish Mathematical Competition, 5

Let $f(n)$ be defined on the positive integers and satisfy: $f(prime) = 1$, $f(ab) = a f(b) + f(a) b$. Show that $f$ is unique and find all $n$ such that $n = f(n)$.

2015 IMO Shortlist, A4

Tags: functional equation , algebra , dorlir ahmeti , imo proposal

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Russian TST 2016, P2

Tags: algebra , functional equation

Prove that a function $f:\mathbb{R}_+\to\mathbb{R}$ satisfies \[f(x+y)-f(x)-f(y)=f\left(\frac{1}{x}+\frac{1}{y}\right)\]if and only if it satisfies $f(xy)=f(x)+f(y)$.

2007 Germany Team Selection Test, 2

Tags: function , functional equation , algebra

Determine all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ which satisfy \[ f \left(\frac {f(x)}{yf(x) \plus{} 1}\right) \equal{} \frac {x}{xf(y)\plus{}1} \quad \forall x,y > 0\]

2008 Canada National Olympiad, 2

Tags: function , algebra , functional equation

Determine all functions $ f$ defined on the set of rational numbers that take rational values for which \[ f(2f(x) \plus{} f(y)) \equal{} 2x \plus{} y, \] for each $ x$ and $ y$.

1985 Traian Lălescu, 1.3

Tags: function , algebra , polynomial , functional equation

Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ with the property that $$ f\left( p(x)\right) =p\left( f(x)\right) ,\quad\forall x\in\mathbb{Q} , $$ for all integer polynomials $ p. $

1994 Abels Math Contest (Norwegian MO), 3b

Tags: functional equation , functional , algebra

Prove that there is no function $f : Z \to Z$ such that $f(f(x)) = x+1$ for all $x$.

2011 Baltic Way, 2

Tags: algebra , functional equation , functional equation in n

Let $f:\mathbb{Z}\to\mathbb{Z}$ be a function such that for all integers $x$ and $y$, the following holds: \[f(f(x)-y)=f(y)-f(f(x)).\] Show that $f$ is bounded.

2023 Turkey Olympic Revenge, 1

Tags: functional equation , function , function in r , algebra

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]

2024 239 Open Mathematical Olympiad, 1

Tags: algebra , functional equation

Let $f:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ be a continuous function such that $f(0)=0$ and $$f(x)+f(f(x))+f(f(f(x)))=3x$$ for all $x>0$. Show that $f(x)=x$ for all $x>0$.

2018 Iran MO (3rd Round), 2

Tags: functional equation , algebra

Find all functions $f: \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that: $f(x^3+xf(xy))=f(xy)+x^2f(x+y) \forall x,y \in \mathbb{R}^{\ge 0}$

2023 Brazil Team Selection Test, 4

Tags: functional equation , fe

Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]

2020 Thailand TSTST, 1

Tags: functional equation , algebra

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y $$ for all $x,y \in \mathbb{R}$.

2023 IMC, 1

Tags: calculus , derivative , functional equation , function

Find all functions $f: \mathbb{R} \to \mathbb{R}$ that have a continuous second derivative and for which the equality $f(7x+1)=49f(x)$ holds for all $x \in \mathbb{R}$.

2010 Dutch IMO TST, 2

Tags: functional equation , maximum , algebra

Find all functions $f : R \to R$ which satisfy $f(x) = max_{y\in R} (2xy - f(y))$ for all $x \in R$.

2017 Dutch IMO TST, 4

Tags: functional equation , algebra

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$ for all $x, y \in \mathbb{R}$.

2010 IMO Shortlist, 6

Tags: positive integer , function , functional equation , algebra

Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$ [i]Proposed by Alex Schreiber, Germany[/i]

2022 Czech-Austrian-Polish-Slovak Match, 2

Tags: algebra , functional equation

Find all functions $f: \mathbb{R^{+}} \rightarrow \mathbb {R^{+}}$ such that $f(f(x)+\frac{y+1}{f(y)})=\frac{1}{f(y)}+x+1$ for all $x, y>0$. [i]Proposed by Dominik Burek, Poland[/i]

2011 Belarus Team Selection Test, 1

Tags: algebra , functional equation , trigonometry , functional

Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$. I.Voronovich

2017 Balkan MO Shortlist, N2

Tags: perfect square , functional equation , functional equation in n , number theory

Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.

2024 Pan-African, 5

Tags: algebra , functional equation

Let $ \mathbb{R} $ denote the set of real numbers. Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that \[ f(x^2) - y f(y) = f(x+y)(f(x) - y) \] for all real numbers $ x $ and $ y $.

2015 Switzerland Team Selection Test, 6

Tags: algebra , polynomial , function , functional equation

Find all polynomial function $P$ of real coefficients such that for all $x \in \mathbb{R}$ $$P(x)P(x+1)=P(x^2+2)$$

2016 Saudi Arabia IMO TST, 3

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ such that $x[f(x + y) - f (x - y)] = 4y f (x)$ for any real numbers $x, y$.

2011 QEDMO 10th, 9

Tags: algebra , functional equation , functional

Let $X = Q-\{-1,0,1\}$. We consider the function $f: X\to X$ given by $f (x) = x -\frac{1}{x} .$ Is there an $a \in X$ such that for every natural number n there is a $y \in X$ with $f (f (...( f (y)) ...)) = a$ where $f$ occurs exactly $n$ times on the left side?

Dumbest FE I ever created, 1.

Tags: function , number theory , functional equation , divisibility , euler s phi function

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$, $$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$