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

2009 Switzerland - Final Round, 9
Tags: inequalities , functional inequality , functional , algebra
Find all injective functions $f : N\to N$ such that holds for all natural numbers $n$: $$f(f(n)) \le \frac{f(n) + n}{2}$$

2013 Chile National Olympiad, 4
Tags: functional , functional equation , number theory , algebra
Consider a function f defined on the positive integers that meets the following conditions: $$f(1) = 1 \, , \,\, f(2n) = 2f(n) \, , \,\, nf(2n + 1) = (2n + 1)(f(n) + n) $$ for all $n \ge 1$. a) Prove that $f(n)$ is an integer for all $n$. b) Find all positive integers $m$ less than $2013$ that satisfy the equation $f(m) = 2m$.

2016 Latvia Baltic Way TST, 4
Tags: algebra , functional , functional equation
Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds: $$f(2^x+2y) =2^yf(f(x))f(y).$$

1976 Chisinau City MO, 130
Tags: algebra , function , functional
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.

VMEO III 2006, 10.3
Tags: algebra , functional
Find all functions $f : R \to R$ that satisfy $f(x^2 + f(y) - y) = (f(x))^2$ for all $x,y \in R$.

2012 Thailand Mathematical Olympiad, 5
Tags: algebra , functional , functional equation
Determine all functions $f : R \to R$ satisfying $f(f(x) + xf(y))= 3f(x) + 4xy$ for all real numbers $x,y$.

2011 Belarus Team Selection Test, 3
Tags: algebra , functional equation , functional
Find all functions $f: R \to R ,g: R \to R$ satisfying the following equality $f(f(x+y))=xf(y)+g(x)$ for all real $x$ and $y$. I. Gorodnin

2001 Kazakhstan National Olympiad, 4
Tags: algebra , functional equation , functional
Find all functions $ f: \mathbb {R} \rightarrow \mathbb {R} $ satisfying the equality $ f (x ^ 2-y ^ 2) = (x-y) (f (x) + f (y)) $ for any $ x, y \in \mathbb {R} $.

1998 Italy TST, 1
Tags: functional equation , functional , function , 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 $x > 0$.

2005 Portugal MO, 6
Tags: algebra , functional , functional equation , number theory
Prove that there is a unique function $f: N\to N$, that verifies $$f(a + b)f(a - b) = f(a^2)$$, for any $a, b\in N$ such that $a > b$.

2022 Harvard-MIT Mathematics Tournament, 7
Tags: algebra , functional
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 $xyz = 1$.

1991 Greece National Olympiad, 1
Tags: function , algebra , functional
Prove that there is no function $f: \mathbb{Z}\to\mathbb{Z}$ such that $f(f(x))=x+1$, for all $x\in\mathbb{Z}$.

1976 Chisinau City MO, 129
Tags: algebra , functional , function , periodic
The function $f (x)$ satisfies the relation $f(x+\pi)=\frac{f(x)}{3f(x) -1}$ for any real number $x$. Prove that the function $f (x)$ is periodic.

2008 Indonesia TST, 4
Tags: number theory , functional equation , functional , functional equation in n
Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies (i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$. (ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.

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

2008 Cuba MO, 4
Tags: algebra , functional
Determine all functions $f : R \to R$ such that $f(xy + f(x)) =xf(y) + f(x)$ for all real numbers $x, y$.

2015 Belarus Team Selection Test, 1
Tags: functional equation , algebra , functional
Do there exist numbers $a,b \in R$ and surjective function $f: R \to R$ such that $f(f(x)) = bx f(x) +a$ for all real $x$? I.Voronovich

1990 Greece National Olympiad, 4
Tags: function , algebra , functional
Find all functions $f: \mathbb{R}^+\to\mathbb{R}$ such that $f(x+y)=f(x^2)+f(y^2)$ for any $x,y \in\mathbb{R}^+$

2010 Saudi Arabia BMO TST, 3
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ such that $$xf(x+xy)= xf(x)+ f(x^2)f(y)$$ for all $x,y \in R$.

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 $x,y \in R$

2004 Switzerland - Final Round, 4
Tags: algebra , functional , functional equation
Determine all functions $f : R \to R$ such that for all $x, y \in R$ holds $$f(xf(x) + f(y)) = y + f(x)^2$$

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

1998 Belarus Team Selection Test, 1
Tags: functional , functional equation , algebra , periodic
Do there exist functions $f : R \to R$ and $g : R \to R$, $g$ being periodic, such that $$x^3= f(\lfloor x \rfloor ) + g(x)$$ for all real $x$ ?

2021-IMOC qualification, A2
Tags: algebra , polynomial , functional equation , functional
Find all functions $f:R \to R$, such that $f(x)+f(y)=f(x+y)$, and there exists non-constant polynomials $P(x)$, $Q(x)$ such that $P(x)f(Q(x))=f(P(x)Q(x))$

2002 Singapore Team Selection Test, 3
Tags: algebra , functional , functional equation
Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$, for all $x \ge 0$.