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

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Found problems: 1513

1981 IMO, 3
Tags: functional equation , function , algebra
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.

VMEO I 2004, 5
Tags: algebra , functional , functional equation
Find all the functions $f:R \to R$ satisfying $$(x + y)(f (x)-f (y)) = f (x^2) - f (y^2),\, \forall x, y \in R$$

2019 India IMO Training Camp, P1
Tags: functional equation , algebra
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 $x,y\in\mathbb{Q}_{>0}$

BIMO 2022, 1
Tags: functional equation , algebra
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x,y$, we have $$f(xf(x)+2y)=f(x)^2+x+2f(y)$$

1998 Italy TST, 1
Tags: functional , function , functional equation , algebra
A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

2003 China Team Selection Test, 2
Tags: function , functional equation , algebra
Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.

2011 IMO Shortlist, 3
Tags: algebra , function , functional equation
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$. [i]Proposed by Japan[/i]

2013 Thailand Mathematical Olympiad, 6
Tags: algebra , functional , functional equation
Determine all functions $f$ : $\mathbb R\to\mathbb R$ satisfying $(x^2+y^2)f(xy)=f(x)f(y)f(x^2+y^2)$ $\forall x,y\in\mathbb R$

2014 BMT Spring, 6
Tags: fe , functional equation , algebra
Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation $$4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1.$$

III Soros Olympiad 1996 - 97 (Russia), 11.2
Tags: algebra , functional equation
Is there a function $f(x)$ defined and continuous on $R$ such that: a) $f(f(x)) = 1 + 2x$ ? b) $f(f(x)) = 1 - 2x $?

2008 IMO, 4
Tags: algebra , functional equation
Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$ [i]Author: Hojoo Lee, South Korea[/i]

2021 Balkan MO Shortlist, A3
Tags: functional equation , algebra
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$. [i]Proposed by Athanasios Kontogeorgis, Greece[/i]

1988 IMO, 3
Tags: binary representation , algebra , function , functional equation
A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

2015 Dutch BxMO/EGMO TST, 5
Tags: algebra , functional equation
Find all functions $f : R \to R$ satisfying $(x^2 + y^2)f(xy) = f(x)f(y)f(x^2 + y^2)$ for all real numbers $x$ and $y$.

2022 Kazakhstan National Olympiad, 3
Tags: algebra , functional equation
Given $m\in\mathbb{N}$. Find all functions $f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}}$ such that $$f(f(x)+y)-f(x)=\left( \frac{f(y)}{y}-1\right)x+f^m(y)$$ holds for all $x,y\in\mathbb{R^{+}}.$ ($f^m(x) =$ $f$ applies $m$ times.)

2013 ELMO Shortlist, 3
Tags: polynomial , algebra , functional equation
Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$. [i]Proposed by Calvin Deng[/i]

2012 Putnam, 3
Tags: topology , algebra , limit , function , functional equation
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function such that (i) $f(x)=\frac{2-x^2}{2}f\left(\frac{x^2}{2-x^2}\right)$ for every $x$ in $[-1,1],$ (ii) $ f(0)=1,$ and (iii) $\lim_{x\to 1^-}\frac{f(x)}{\sqrt{1-x}}$ exists and is finite. Prove that $f$ is unique, and express $f(x)$ in closed form.

2001 IMC, 5
Tags: function , functional equation
Prove that there is no function $f: \mathbb{R} \rightarrow \mathbb{R}$ with $f(0) >0$, and such that \[f(x+y) \geq f(x) +yf(f(x)) \text{ for all } x,y \in \mathbb{R}. \]

2012 IMO Shortlist, A5
Tags: function , algebra , functional equation
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions \[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\] and $f(-1) \neq 0$.

2023 IRN-SGP-TWN Friendly Math Competition, 6
Tags: polynomial with integer coeffi , algebra , number theory , functional equation , polynomial
$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

2016 India IMO Training Camp, 2
Tags: function , algebra , functional equation
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)

2009 Thailand Mathematical Olympiad, 5
Tags: functional , algebra , functional equation
Determine all functions $f : R\to R$ satisfying: $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$ for all real numbers $x, y$.

2015 Postal Coaching, Problem 1
Tags: functional equation , algebra
Let $f:\mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}$ be defined by $f(0)=0$, $$f(2n+1)=2f(n)$$ for $n \ge 0$ and $$f(2n)=2f(n)+1$$ for $n \ge 1$ If $g(n)=f(f(n))$, prove that $g(n-g(n))=0$ for all $n \ge 0$.

1997 All-Russian Olympiad Regional Round, 11.8
Tags: functional , algebra , functional equation
For which $a$, there is a function $f: R \to R$, different from a constant, such that $$f(a(x + y)) = f(x) + f(y) ?$$

2016 District Olympiad, 3
Tags: function , continuity , real analysis , algebra , functional equation
Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property: $$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$