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

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

1981 Poland - Second Round, 3
Tags: algebra , continuous , functional
Prove that there is no continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the condition $ f(f(x)) = - x $ for every $ x $.

2015 Costa Rica - Final Round, 5
Tags: algebra , inequalities , functional inequality , functional
Let $f: N^+ \to N^+$ be a function that satisfies that $$kf(n) \le f (kn) \le kf(n)+ k- 1, \,\, \forall k,n \in N^+$$ Prove that $$f(a) + f(b) \le f (a + b) \le f(a) + f(b) + 1, \,\, \forall a, b \in N^+$$

2010 Saudi Arabia IMO TST, 3
Tags: function , functional equation , functional , algebra
Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$, for all $n \in N$. Find $f(2010)$. Note: $N = \{0,1,2,...\}$

2009 Cuba MO, 4
Tags: algebra , functional
Determine all the functions $f : R \to R$ such that: $$x + f(xf(y)) = f(y) + yf(x)$$ for all $x, y \in R$.

2000 Swedish Mathematical Competition, 5
Tags: functional equation , functional , number theory
Let $f(n)$ be defined on the positive integers and satisfy: $f(prime) = 1$, $f(ab) = a f(b) + f(a) b$. Show that $f$ is unique and find all $n$ such that $n = f(n)$.

1994 Abels Math Contest (Norwegian MO), 3b
Tags: functional equation , functional , algebra
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, 1
Tags: algebra , functional equation , trigonometry , functional
Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$. I.Voronovich

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

2011 QEDMO 10th, 9
Tags: algebra , functional equation , functional
Let $X = Q-\{-1,0,1\}$. We consider the function $f: X\to X$ given by $f (x) = x -\frac{1}{x} .$ Is there an $a \in X$ such that for every natural number n there is a $y \in X$ with $f (f (...( f (y)) ...)) = a$ where $f$ occurs exactly $n$ times on the left side?

VI Soros Olympiad 1999 - 2000 (Russia), 9.4
Tags: algebra , functional , functional inequation , inequalities
Is there a function $f(x)$, which satisfies both of the following conditions: a) if $x \ne y$, then $f(x)\ne f(y)$ b) for all real $x$, holds the inequality $f(x^2-1998x)-f^2(2x-1999)\ge \frac14$?

2015 Costa Rica - Final Round, 3
Tags: algebra , fractional part , functional equation , functional
Indicate (justifying your answer) if there exists a function $f: R \to R$ such that for all $x \in R$ fulfills that i) $\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x)$ ii) $f (f(x)) = f(x)$ where $\{m\}$ denotes the fractional part of $m$. That is, $\{2.657\} = 0.657$, and $\{-1.75\} = 0.25$.

2018 NZMOC Camp Selection Problems, 10
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$ for all $x, y \in R$.

2019 Abels Math Contest (Norwegian MO) Final, 3b
Tags: function , functional equation , functional , algebra
Find all real functions $f$ defined on the real numbers except zero, satisfying $f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$

2019 Greece Team Selection Test, 4
Tags: functional equation , functional , algebra
Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.

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

2017 Saudi Arabia BMO TST, 2
Tags: function , functional inequality , functional , algebra
Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied: i) $2f (x) + 2f (y) \le f (x + y)$ ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$

2019 Saudi Arabia Pre-TST + Training Tests, 1.3
Tags: functional , algebra
Find all functions $f : R^+ \to R^+$ such that $f(3 (f (xy))^2 + (xy)^2) = (xf (y) + yf (x))^2$ for any $x, y > 0$.

1987 Austrian-Polish Competition, 3
Tags: sequence , functional , functional equation , limit
A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.

2015 Estonia Team Selection Test, 5
Tags: algebra , functional equation , functional
Find all functions $f$ from reals to reals which satisfy $f (f(x) + f(y)) = f(x^2) + 2x^2 f(y) + (f(y))^2$ for all real numbers $x$ and $y$.

2013 Grand Duchy of Lithuania, 1
Tags: functional , increasing , integer
Let $f : R \to R$ and $g : R \to R$ be strictly increasing linear functions such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that $f(x) - g(x)$ is an integer for any $x \in R$.

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 $b$.

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 $x, y$.

1992 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: number theory , functional , functional equation
Let $f:Z \to N -\{0\}$ such that: $f(x + y)f(x-y) = (f(x)f(y))^2$ and $f(1)\ne 1$. Provethat $\log_{f(1)}f(z)$ is a perfect square for every integer $z$.

2018 Saudi Arabia IMO TST, 1
Tags: number theory , divide , divisible , functional
Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.