Olympiad problems

2001 IMO Shortlist, 4

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, satisfying \[ f(xy)(f(x) - f(y)) = (x-y)f(x)f(y) \] for all $x,y$.

2016 Romania Team Selection Tests, 2

Tags: function , number theory , functional equation , algebra

Determine all $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ such that $f(m)\geq m$ and $f(m+n) \mid f(m)+f(n)$ for all $m,n\in \mathbb{Z}^+$

2011 ELMO Shortlist, 2

Tags: function , induction , functional equation , algebra

Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$, \[f(a+d)+f(b-c)=f(a-d)+f(b+c).\] [i]Calvin Deng.[/i]

PEN K Problems, 32

Tags: functional equation , function

Find all functions $f: \mathbb{Z}^{2}\to \mathbb{R}^{+}$ such that for all $i, j \in \mathbb{Z}$: \[f(i,j)=\frac{f(i+1, j)+f(i,j+1)+f(i-1,j)+f(i,j-1)}{4}.\]

2019 Teodor Topan, 3

Tags: function , algebra , functional equation , find all functions

Let be a positive real number $ r, $ a natural number $ n, $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying $ f(rxy)=(f(x)f(y))^n, $ for any real numbers $ x,y. $ [b]a)[/b] Give three distinct examples of what $ f $ could be if $ n=1. $ [b]b)[/b] For a fixed $ n\ge 2, $ find all possibilities of what $ f $ could be. [i]Bogdan Blaga[/i]

1991 Putnam, B2

Tags: function , functional equation , algebra , superior algebra

Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$. If $f(x+y)=f(x)f(y)-g(x)g(y)$, $g(x+y)=f(x)g(y)+g(x)f(y)$, and $f'(0)=0$, prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$.

2018 Czech-Polish-Slovak Match, 1

Tags: function , functional equation , algebra

Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$ [i]Proposed by Walther Janous, Austria[/i]

1994 IMO Shortlist, 5

Tags: function , induction , functional equation , algebra

Let $ f(x) \equal{} \frac{x^2\plus{}1}{2x}$ for $ x \neq 0.$ Define $ f^{(0)}(x) \equal{} x$ and $ f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x))$ for all positive integers $ n$ and $ x \neq 0.$ Prove that for all non-negative integers $ n$ and $ x \neq \{\minus{}1,0,1\}$ \[ \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.\]

PEN K Problems, 31

Tags: function , induction , functional equation

Find all strictly increasing functions $f: \mathbb{N}\to \mathbb{N}$ such that \[f(f(n))=3n.\]

2019 Germany Team Selection Test, 1

Tags: functional equation , algebra

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

2022 Korea Junior Math Olympiad, 4

Tags: algebra , functional equation , function

Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.

2021 Francophone Mathematical Olympiad, 4

Tags: number theory , greatest common divisor , least common multiple , functional equation , functional

Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$: \[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) = \mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\] Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$, and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$.

2024 Mongolian Mathematical Olympiad, 3

Tags: functional equation , function , algebra

Let $\mathbb{R}^+$ denote the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all positive real numbers $x$ and $y$ : \[f(x)f(y+f(x))=f(1+xy)\] [i]Proposed by Otgonbayar Uuye. [/i]

2008 Chile National Olympiad, 3

Tags: algebra , functional equation , function

Determine all strictly increasing functions $f : R \to R$ such that for all $x \ne y$ to hold $$\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}$$

2024 India IMOTC, 23

Tags: number theory , functional equation

Prove that there exists a function $f : \mathbb{N} \mapsto \mathbb{N}$ that satisfies the following: [color=#FFFFFF]___[/color]1. For all positive integers $m, n$ we have \[\gcd(|f(m)-f(n)|, f(mn)) = f(\gcd(m, n))\] [color=#FFFFFF]___[/color]2. For all positive integers $m$, we have $f(f(m)) = f(m)$. [color=#FFFFFF]___[/color]3. For all positive integers $k$, there exists a positive integer $n$ with $2024^{k} \mid f(n)$. [i]Proposed by MV Adhitya, Archit Manas[/i]

2007 Macedonia National Olympiad, 4

Tags: function , functional equation , algebra

Find all functions $ f : \mathbb{R}\to\mathbb{R}$ that satisfy \[ f (x^{3} \plus{} y^{3}) \equal{} x^{2}f (x) \plus{} yf (y^{2}) \] for all $ x, y \in\mathbb R.$

1996 Korea National Olympiad, 2

Tags: functional equation , algebra , number theory , function

Let the $f:\mathbb{N}\rightarrow\mathbb{N}$ be the function such that (i) For all positive integers $n,$ $f(n+f(n))=f(n)$ (ii) $f(n_o)=1$ for some $n_0$ Prove that $f(n)\equiv 1.$

2019 Nigerian Senior MO Round 4, 1

Tags: function , functional equation , number theory , inequalities

Let $f: N \to N$ be a function satisfying (a) $1\le f(x)-x \le 2019$ $\forall x \in N$ (b) $f(f(x))\equiv x$ (mod $2019$) $\forall x \in N$ Show that $\exists x \in N$ such that $f^k(x)=x+2019 k, \forall k \in N$

2023 Dutch IMO TST, 4

Tags: functional equation , functional equation in r , algebra

Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.

2008 Postal Coaching, 4

Tags: functional , functional equation , algebra

Find all functions $f : R \to R$ such that $$f(xf(y))= (1 - y)f(xy) + x^2y^2f(y)$$ for all reals $x, y$.

1981 Romania Team Selection Tests, 4.

Tags: algebra , functional equation

Determine the function $f:\mathbb{R}\to\mathbb{R}$ such that $\forall x\in\mathbb{R}$ \[f(x)+f(\lfloor x\rfloor)f(\{x\})=x,\] and draw its graph. Find all $k\in\mathbb{R}$ for which the equation $f(x)+mx+k=0$ has solutions for any $m\in\mathbb{R}$. [i]V. Preda and P. Hamburg[/i]