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

2007 Cuba MO, 4
Tags: functional , functional equation , algebra
Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$ for all positive real $x, y$.

2016 Switzerland - Final Round, 10
Tags: functional , functional equation , algebra
Find all functions $f : R \to R$ such that for all $x, y \in R$: $$f(x + yf(x + y)) = y^2 + f(xf(y + 1)).$$

2017 Puerto Rico Team Selection Test, 1
Tags: functional , functional equation , sum , algebra
Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$. Calculate $\sum_{k = 1}^{10}f (k!)$.

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

MathLinks Contest 3rd, 2
Tags: algebra , functional equation , functional , number theory
Find all functions $f : \{1, 2, ... , n,...\} \to Z$ with the following properties (i) if $a, b$ are positive integers and $a | b$, then $f(a) \ge f(b)$; (ii) if $a, b$ are positive integers then $f(ab) + f(a^2 + b^2) = f(a) + f(b)$.

2019 Saudi Arabia Pre-TST + Training Tests, 4.2
Tags: functional , algebra , functional equation
Find all functions $f : R^2 \to R$ that for all real numbers $x, y, z$ satisfies to the equation $f(f(x,z), f(z, y))= f(x, y) + z$

2012 Cuba MO, 7
Tags: algebra , functional equation , functional
Find all the functions $f : R\to R$ such that $f(x^2 + f(y)) = y - x^2$ for all $x, y$ reals.

2016 Saudi Arabia GMO TST, 2
Tags: algebra , functional , functional equation
Find all functions $f : Z \to Z$ such that $f (2m + f (m) + f (m)f (n)) = nf (m) + m$ for any integers $m, n$

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 $y$.

1984 Austrian-Polish Competition, 9
Tags: algebra , functional equation , functional
Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$

2015 Saudi Arabia GMO TST, 1
Tags: functional , algebra , functional equation
Find all functions $f : R \to R$ satisfying the following conditions (a) $f(1) = 1$, (b) $f(x + y) = f(x) + f(y)$, $\forall (x,y) \in R^2$ (c) $f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }$, $\forall x \in R -\{0\}$ Trần Nam Dũng

1989 Romania Team Selection Test, 1
Tags: algebra , functional , functional equation
Let $F$ be the set of all functions $f : N \to N$ which satisfy $f(f(x))-2 f(x)+x = 0$ for all $x \in N$. Determine the set $A =\{ f(1989) | f \in F\}$.

2008 Indonesia TST, 4
Tags: functional equation , functional , functional equation in n , number theory
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 $f(p)|f(n)$.

2018 Estonia Team Selection Test, 4
Tags: algebra , functional , functional equation
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$

VI Soros Olympiad 1999 - 2000 (Russia), 10.3
Tags: algebra , functional equation , functional
Find all functions $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions: 1) $|f(x)|\ge 1$, 2) $f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real values of $x $ and $y$.

1996 Singapore Team Selection Test, 2
Tags: functional equation , functional , algebra
Prove that there is a function $f$ from the set of all natural numbers to itself such that for any natural number $n$, $f(f(n)) = n^2$.

2008 Indonesia TST, 2
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.

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

2006 Cuba MO, 4
Tags: functional , algebra , function
Let $f : Z_+ \to Z_+$ such that: a) $f(n + 1) > f(n)$ for all $n \in Z_+$ b) $f(n + f(m)) = f(n) + m + 1$ for all $n,m \in Z_+$ Find $f(2006)$.

2006 Thailand Mathematical Olympiad, 5
Tags: functional , algebra , function , functional equation
Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$.

2005 Switzerland - Final Round, 9
Tags: functional , algebra , functional equation
Find all functions $f : R^+ \to R^+$ such that for all $x, y > 0$ $$f(yf(x))(x + y) = x^2(f(x) + f(y)).$$

1991 Austrian-Polish Competition, 9
Tags: functional , algebra , composition , function
For a positive integer $n$ denote $A = \{1,2,..., n\}$. Suppose that $g : A\to A$ is a fixed function with $g(k) \ne k$ and $g(g(k)) = k$ for $k \in A$. How many functions $f: A \to A$ are there such that $f(k)\ne g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$?

2000 Swedish Mathematical Competition, 5
Tags: functional , functional equation , 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 $n = f(n)$.

2020 Nordic, 4
Tags: functional , algebra , functional equation
Find all functions $f : R- \{-1\} \to R$ such that $$f(x)f \left( f \left(\frac{1 - y}{1 + y} \right)\right) = f\left(\frac{x + y}{xy + 1}\right) $$ for all $x, y \in R$ that satisfy $(x + 1)(y + 1)(xy + 1) \ne 0$.

1998 Switzerland Team Selection Test, 10
Tags: periodic , functional , function
5. Let $f : R \to R$ be a function that satisfies for all $x \in R$ (i) $| f(x)| \le 1$, and (ii) $f\left(x+\frac{13}{42}\right)+ f(x) = f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right)$ Prove that $f$ is a periodic function