Olympiad problems

2012 Baltic Way, 5

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ for which \[f(x + y) = f(x - y) + f(f(1 - xy))\] holds for all real numbers $x$ and $y$.

1979 IMO Longlists, 80

Tags: functional equation , algebra

Prove that the functional equations \[f(x + y) = f(x) + f(y),\] \[ \text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)\] are equivalent.

2015 Indonesia MO Shortlist, A6

Tags: algebra , functional equation , function

Let functions $f, g: \mathbb{R}^+ \to \mathbb{R}^+$ satisfy the following: \[ f(g(x)y + f(x)) = (y+2015)f(x) \] for every $x,y \in \mathbb{R}^+$. (a) Prove that $g(x) = \frac{f(x)}{2015}$ for every $x \in \mathbb{R}^+. $ (b) State an example of function that satisfy the equation above and $f(x), g(x) \ge 1$ for every $x \in \mathbb{R}^+$.

1993 All-Russian Olympiad, 3

Tags: function , domain , functional equation , 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 $f(x^y)=f(x)^{f(y)}$.

2013 Thailand Mathematical Olympiad, 6

Tags: functional equation , functional , algebra

Determine all functions $f$ : $\mathbb R\to\mathbb R$ satisfying $(x^2+y^2)f(xy)=f(x)f(y)f(x^2+y^2)$ $\forall x,y\in\mathbb R$

2021 Serbia National Math Olympiad, 5

Tags: functional equation , function , algebra

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for every $x,y\in\mathbb{R}$ the following equality holds: $$f(xf(y)+x^2+y)=f(x)f(y)+xf(x)+f(y).$$

STEMS 2021-22 Math Cat A-B, A3 B1

Tags: algebra , functional equation

Find all functions $f :\mathbb{N} \rightarrow \mathbb{N}$ such that $f(m + f(n)f(m)) = nf(m) + m$ holds for all $m,n \in \mathbb{N}$.

1992 IMO Longlists, 24

Tags: function , algebra , functional equation

[i](a)[/i] Show that there exists exactly one function $ f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions: [b](i)[/b] if $0 < q < \frac 12$, then $f(q)=1+f \left( \frac{q}{1-2q} \right);$ [b](ii)[/b] if $1 < q \leq 2$, then $f(q) = 1+f(q + 1);$ [b](iii)[/b] $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$ [i](b)[/i] Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$

2005 Slovenia Team Selection Test, 2

Tags: functional , functional equation , algebra

Find all functions $f : R^+ \to R^+$ such that $x^2(f(x)+ f(y)) = (x+y)f (f(x)y)$ for any $x,y > 0$.

2007 Germany Team Selection Test, 2

Tags: function , functional equation , algebra

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

2023 Mexican Girls' Contest, 4

Tags: functional equation , algebra

A function $g$ is such that for all integer $n$: $$g(n)=\begin{cases} 1\hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\geq 1 & \\ 0 \hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\leq 0 & \end{cases}$$ A function $f$ is such that for all integers $n\geq 0$ and $m\geq 0$: $$f(0,m)=0 \hspace{0.5cm} \textrm{and}$$ $$f(n+1,m)=\Bigl(1-g(m)+g(m)\cdot g(m-1-f(n,m))\Bigr)\cdot\Bigl(1+f(n,m)\Bigr)$$ Find all the possible functions $f(m,n)$ that satisfies the above for all integers $n\geq0$ and $m\geq 0$

2014 Brazil Team Selection Test, 1

Tags: number theory , algebra , functional equation

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 $n$.

1996 Estonia Team Selection Test, 3

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$: $(i)$ $f(x)=-f(-x);$ $(ii)$ $f(x+1)=f(x)+1;$ $(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$

VMEO III 2006 Shortlist, A1

Tags: algebra , functional equation , functional

Find all functions $f:R \to R$ such that $$f(x^2+f(y)-y) =(f(x))^2-f(y)$$ for all $x,y \in R$

2021 2nd Memorial "Aleksandar Blazhevski-Cane", 6

Tags: algebra , functional equation

Let $\mathbb{R}^{+}$ be the set of all positive real numbers. Find all the functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for all $x, y \in \mathbb{R}^{+}$, \[ f(x)f(y) = f(y)f(xf(y)) + \frac{1}{xy}. \]

2020 Israel Olympic Revenge, P1

Tags: functional equation , algebra , function , symmetry

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$ one has \[f(f(x)+y)=f(x+f(y))\] and in addition the set $f^{-1}(a)=\{b\in \mathbb{R}\mid f(b)=a\}$ is a finite set for all $a\in \mathbb{R}$.

1978 IMO, 3

Tags: algebra , functional equation , sequence , partition

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

2014 MMATHS, 3

Tags: algebra , functional equation

Let $f : R^+ \to R^+$ be a function satisfying $$f(\sqrt{x_1x_2}) =\sqrt{f(x_1)f(x_2)}$$ for all positive real numbers $x_1, x_2$. Show that $$f( \sqrt[n]{x_1x_2... x_n}) = \sqrt[n]{f(x_1)f(x_2) ... f(x_n)}$$ for all positive integers $n$ and positive real numbers $x_1, x_2,..., x_n$.

1998 IMO Shortlist, 5

Tags: number theory , prime number , algebra , functional equation

Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

1993 Czech And Slovak Olympiad IIIA, 5

Tags: functional equation , functional , algebra

Find all functions $f : Z \to Z$ such that $f(-1) = f(1)$ and $f(x)+ f(y) = f(x+2xy)+ f(y-2xy)$ for all $x,y \in Z$

2019 Latvia Baltic Way TST, 2

Tags: function , functional equation , algebra

Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that $$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$ holds for all real numbers $x; y$

2019 Thailand TSTST, 3

Tags: functional equation , algebra , function

Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $\text{(i)}$ $f(f(m)+n)+2m=f(n)+f(3m)$ for every $m,n\in\mathbb{Z}$, $\text{(ii)}$ there exists a $d\in\mathbb{Z}$ such that $f(d)-f(0)=2$, and $\text{(iii)}$ $f(1)-f(0)$ is even.

2010 Philippine MO, 3

Tags: function , functional equation , algebra

Let $\mathbb{R}^*$ be the set of all real numbers, except $1$. Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010$$.

2019 Thailand TST, 1

Tags: algebra , functional equation

Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

2022 EGMO, 2

Tags: number theory , functional equation

Let $\mathbb{N}=\{1, 2, 3, \dots\}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any positive integers $a$ and $b$, the following two conditions hold: (1) $f(ab) = f(a)f(b)$, and (2) at least two of the numbers $f(a)$, $f(b)$, and $f(a+b)$ are equal.