Olympiad problems

2015 Saudi Arabia IMO TST, 1

Find all functions $f : R_{>0} \to R$ such that $f \left(\frac{x}{y}\right) = f(x) + f(y) - f(x)f(y)$ for all $x, y \in R_{>0}$. Here, $R_{>0}$ denotes the set of all positive real numbers. Nguyễn Duy Thái Sơn

2020 Estonia Team Selection Test, 3

Tags: algebra , functional , functional equation

Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$

2025 Taiwan TST Round 2, A

Tags: algebra , functional equation

Find all $g:\mathbb{R}\to\mathbb{R}$ so that there exists a unique $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)$ and \[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\] for all $x,y\in\mathbb{R}$. [i] Proposed by usjl[/i]

2012 IMAC Arhimede, 3

Tags: algebra , functional equation , functional equation in q

Find all functions $f:Q^+ \to Q^+$ such that for any $x,y \in Q^+$ : $$y=\frac{1}{2}\left[f\left(x+\frac{y}{x}\right)- \left(f(x)+\frac{f(y)}{f(x)}\right)\right]$$

2021 Thailand TST, 3

Tags: number theory , functional equation , nonnegative integers

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

2021 Taiwan TST Round 3, 4

Tags: algebra , functional equation , iteration

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying \[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\] for all integers $a$ and $b$

2023 Philippine MO, 8

Tags: geometry , functional equation , combinatorial geometry , function , nice

Let $\mathcal{S}$ be the set of all points in the plane. Find all functions $f : \mathcal{S} \rightarrow \mathbb{R}$ such that for all nondegenerate triangles $ABC$ with orthocenter $H$, if $f(A) \leq f(B) \leq f(C)$, then $$f(A) + f(C) = f(B) + f(H).$$

2021 Israel Olympic Revenge, 1

Tags: number theory , functional equation , algebra , function

Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to\mathbb N$ such that $$\frac{f(x)-f(y)+x+y}{x-y+1}$$ is an integer, for all positive integers $x,y$ with $x>y$.

2015 IFYM, Sozopol, 1

Tags: functional equation , algebra

Determine all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy the following equations: a) $f(f(n))=4n+3$ $\forall$ $n \in \mathbb{Z}$; b) $f(f(n)-n)=2n+3$ $\forall$ $n \in \mathbb{Z}$.

2021-IMOC, A8

Tags: functional equation , algebra , inequality , function , inequalities

Find all functions $f : \mathbb{N} \to \mathbb{N}$ with $$f(x) + yf(f(x)) < x(1 + f(y)) + 2021$$ holds for all positive integers $x,y.$

2018 Hanoi Open Mathematics Competitions, 2

Tags: polynomial , functional equation

Let $f(x)$ be a polynomial such that $2f(x) + f(2 - x) = 5 + x$ for any real number x. Find the value of $f(0) + f(2)$. A. $4$ B. $0$ C.$ 2$ D. $3$ E. $1$

2005 iTest, 4

Tags: function , functional equation , functional , algebra

The function f is defined on the set of integers and satisfies $\bullet$ $f(n) = n - 2$, if $n \ge 2005$ $\bullet$ $f(n) = f(f(n+7))$, if $n < 2005$. Find $f(3)$.

2023 APMO, 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 \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]

2014 Thailand Mathematical Olympiad, 2

Tags: function , algebra , functional equation

Find all functions $f : R \to R$ satisfying $f(xy - 1) + f(x)f(y) = 2xy - 1$ for all real numbers $x, y$

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 $f(x^y)=f(x)^{f(y)}$.

1992 IMO Longlists, 51

Tags: algebra , polynomial , functional equation , number theory

Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose \[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\] Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$

2014 Middle European Mathematical Olympiad, 1

Tags: function , algebra , functional equation , fe , real

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

2011 QEDMO 9th, 7

Tags: functional equation , functional , algebra

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

2016 Korea Winter Program Practice Test, 3

Tags: functional equation , algebra

Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(y)+yf(z)+zf(x))=yf(x)+zf(y)+xf(z)$

2021 Balkan MO Shortlist, A6

Tags: algebra , functional equation

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

2015 Indonesia MO, 4

Tags: function , algebra , functional equation

Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies \[ f(g(x)y + f(x)) = (y+2015)f(x) \] for every $x,y \in \mathbb{R^+} $ a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$ b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$

2001 Dutch Mathematical Olympiad, 2

Tags: functional equation , algebra

The function f has the following properties : $f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$ $f(4) = 10$ Calculate $f(2001)$.

2015 Saudi Arabia GMO TST, 1

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ satisfying the following conditions (a) $f(1) = 1$, (b) $f(x + y) = f(x) + f(y)$, $\forall (x,y) \in R^2$ (c) $f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }$, $\forall x \in R -\{0\}$ Trần Nam Dũng

2020 Iran Team Selection Test, 4

Tags: functional equation , algebra

Given a function $g:[0,1] \to \mathbb{R}$ satisfying the property that for every non empty dissection of the trivial $[0,1]$ to subsets $A,B$ we have either $\exists x \in A; g(x) \in B$ or $\exists x \in B; g(x) \in A$ and we have furthermore $g(x)>x$ for $x \in [0,1]$. Prove that there exist infinite $x \in [0,1]$ with $g(x)=1$. [i]Proposed by Ali Zamani [/i]

2006 Federal Competition For Advanced Students, Part 2, 2

Tags: function , functional equation , algebra

Find all monotonous functions $ f: \mathbb{R} \to \mathbb{R}$ that satisfy the following functional equation: \[f(f(x)) \equal{} f( \minus{} f(x)) \equal{} f(x)^2.\]