This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 313

2012 Grand Duchy of Lithuania, 1
Tags: functional , functional equation , algebra
Find all functions $g : R \to R$, for which there exists a strictly increasing function $f : R \to R$ such that $f(x + y) = f(x)g(y) + f(y)$.

2001 Estonia Team Selection Test, 3
Tags: functional , functional equation , algebra
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

1994 Abels Math Contest (Norwegian MO), 3b
Tags: algebra , functional , functional equation
Prove that there is no function $f : Z \to Z$ such that $f(f(x)) = x+1$ for all $x$.

2011 Belarus Team Selection Test, 3
Tags: functional , algebra , functional equation
Find all functions $f:R\to R$ such that for all real $x,y$ with $y\ne 0$ $$f(x-f(x/y))=xf(1-f(1/y))$$ and a) $f(1-f(1))\ne 0$ b) $ f(1-f(1))= 0$ S. Kuzmich, I.Voronovich

2022 Auckland Mathematical Olympiad, 9
Tags: functional , algebra , functional equation , number theory
Does there exist a function $f(n)$, which maps the set of natural numbers into itself and such that for each natural number $n > 1$ the following equation is satisfi ed $$f(n) = f(f(n - 1)) + f(f(n + 1))?$$

1986 Dutch Mathematical Olympiad, 1
Tags: functional , algebra
$f(x) = \frac{12x+9}{19x+86}, \,\, x \ne -\frac{86}{19}$ Prove that $\exists ! \,\,\, {x_o \in R} \,\,\, \forall h_1,h_2 \in R [f(x_0+h_1)f(x_0-h_1)=f(x_0+h_2)f(x_0-h_2)]$, and calculate $x_0$.

1988 Austrian-Polish Competition, 4
Tags: algebra , functional equation , functional , increasing
Determine all strictly increasing functions $f: R \to R$ satisfying $f (f(x) + y) = f(x + y) + f (0)$ for all $x,y \in R$.

1995 Grosman Memorial Mathematical Olympiad, 6
Tags: functional , algebra , functional equation
(a) Prove that there is a unique function $f : Q \to Q$ satisfying: (i) $f(q)= 1 + f\left(\frac{q}{1-2q}\right)$ for $0<q< \frac12$ (ii) $f(q)= 1 + f(q-1)$ for $1<q\le 2$ (iii) $f(q)f\left(\frac{1}{q}\right)=1$ for all $q\in Q^+$ (b) For this function $f$ , find all $r\in Q^+$ such that $f(r) = r$

1973 Swedish Mathematical Competition, 6
Tags: functional , algebra , functional inequality
$f(x)$ is a real valued function defined for $x \geq 0$ such that $f(0) = 0$, $f(x+1)=f(x)+\sqrt{x}$ for all $x$, and \[ f(x) < \frac{1}{2}f\left(x - \frac{1}{2}\right)+\frac{1}{2}f\left(x + \frac{1}{2}\right) \quad \text{for all} \quad x \geq \frac{1}{2} \] Show that $f\left(\frac{1}{2}\right)$ is uniquely determined.

2013 Switzerland - Final Round, 4
Tags: functional , functional equation , algebra
Find all functions $f : R_{>0} \to R_{>0}$ with the following property: $$f \left( \frac{x}{y + 1}\right) = 1 - xf(x + y)$$ for all $x > y > 0$ .

1989 Chile National Olympiad, 6
Tags: functional , functional equation , function , algebra
The function $f$, with domain on the set of non-negative integers, is defined by the following : $\bullet$ $f (0) = 2$ $\bullet$ $(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4$, taking $f (n)$ the largest possible value. Determine $f (n)$.

2012 Swedish Mathematical Competition, 1
Tags: functional , functional equation , algebra
The function $f$ satisfies the condition $$f (x + 1) = \frac{1 + f (x)}{1 - f (x)}$$ for all real $x$, for which the function is defined. Determine $f(2012)$, if we known that $f(1000)=2012$.

2020 Dutch BxMO TST, 3
Tags: functional , functional equation , algebra
Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$

2011 Abels Math Contest (Norwegian MO), 3b
Tags: functional , algebra , function , functional inequalities , inequalities
Find all functions $f$ from the real numbers to the real numbers such that $f(xy) \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$.

2009 Thailand Mathematical Olympiad, 2
Tags: functional , functional equation , function , algebra , injective
Is there an injective function $f : Z^+ \to Q$ satisfying the equation $f(xy) = f(x) + f(y)$ for all positive integers $x$ and $y$?

2016 Saudi Arabia IMO TST, 3
Tags: functional , algebra , 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 $x, y$.

1993 Czech And Slovak Olympiad IIIA, 5
Tags: algebra , functional equation , functional
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 $x,y \in Z$

2016 Estonia Team Selection Test, 3
Tags: functional , algebra , functional equation
Find all functions $f : R \to R$ satisfying the equality $f (2^x + 2y) =2^y f ( f (x)) f (y) $for every $x, y \in R$.

2022 Harvard-MIT Mathematics Tournament, 7
Tags: functional , algebra
Find, with proof, all functions $f : R - \{0\} \to R$ such that $$f(x)^2 - f(y)f(z) = x(x + y + z)(f(x) + f(y) + f(z))$$ for all real $x, y, z$ such that $xyz = 1$.

2010 Saudi Arabia BMO TST, 4
Tags: number theory , functional , function , sequence , algebra
Let $f : N \to [0, \infty)$ be a function satisfying the following conditions: a) $f(4)=2$ b) $\frac{1}{f( 0 ) + f( 1)} + \frac{1}{f( 1 ) + f( 2 )} + ... + \frac{1}{f (n ) + f(n + 1) }= f ( n + 1)$ for all integers $n \ge 0$. Find $f(n)$ in closed form.

2003 Olympic Revenge, 6
Tags: algebra , functional , functional equation
Find all functions $f:R^{*} \rightarrow R$ such that $f(x)\not = x$ and $$ f(y(f(x)-x))=\frac{f(x)}{y}-\frac{f(y)}{x} $$ for any $x,y \not = 0$.

2015 Saudi Arabia BMO TST, 1
Tags: functional , functional equation , increasing , function , algebra
Find all strictly increasing functions $f : Z \to R$ such that for any $m, n \in Z$ there exists a $k \in Z$ such that $f(k) = f(m) - f(n)$. Nguyễn Duy Thái Sơn

1986 Austrian-Polish Competition, 9
Tags: functional , algebra , continuous , composition , functional equation
Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.

2000 All-Russian Olympiad Regional Round, 10.5
Tags: functional , trigonometry , algebra , inequalities , function
Is there a function $f(x)$ defined for all $x \in R$ and for all $x, y \in R $ satisfying the inequality $$|f(x + y) + \sin x + \sin y| < 2?$$

1976 Chisinau City MO, 130
Tags: algebra , functional , function
Prove that the function $f (x)$ satisfying the relation $|f (x) - f (y) | \le | x - y|^a$ for any real numbers $x, y$ and some number $a> 1$ is constant.