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.

AND:
OR:
NO:

Found problems: 313

1994 Italy TST, 3
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.

1999 Switzerland Team Selection Test, 3
Tags: functional equation , functional , algebra
Find all functions $f : R -\{0\} \to R$ that satisfy $\frac{1}{x}f(-x)+ f\left(\frac{1}{x}\right)= x$ for all $x \ne 0$.

2006 QEDMO 2nd, 13
Tags: algebra , functional , functional equation
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any two reals $x$ and $y$, we have $f\left( f\left( x+y\right) \right) +xy=f\left( x+y\right) +f\left( x\right) f\left( y\right) $.

2012 CHMMC Fall, 2
Tags: algebra , functional , continuous function
Find all continuous functions $f : R \to R$ such that $$f(x + f(y)) = f(x + y) + y,$$ for all $x, y \in R$. No proof is required for this problem.

2006 Cuba MO, 4
Tags: algebra , function , functional
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 $f(2006)$.

1986 Tournament Of Towns, (116) 4
Tags: algebra , continuous , analysis , function , functional equation , functional
The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous. (A.I. Plotkin, Leningrad)

2010 Thailand Mathematical Olympiad, 6
Tags: algebra , functional equation , functional
Let $f : R \to R$ be a function satisfying the functional equation $f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)$ for all reals $x, y$. Show that $f$ is even, that is, $f(-x) = f(x)$ for all reals $x$

2021 Flanders Math Olympiad, 4
Tags: inequalities , functional , algebra , functional equation
(a) Prove that for every $x \in R$ holds that $$-1 \le \frac{x}{x^2 + x + 1} \le \frac 13$$ (b) Determine all functions $f : R \to R$ for which for every $x \in R$ holds that $$f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}$$

2013 Costa Rica - Final Round, F2
Tags: functional equation , functional , algebra
Find all functions $f:R -\{0,2\} \to R$ that satisfy for all $x \ne 0,2$ $$f(x) \cdot \left(f\left(\sqrt[3]{\frac{2+x}{2-x}}\right) \right)^2=\frac{x^3}{4}$$

1998 Slovenia Team Selection Test, 1
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ that satisfy $f((x-y)^2)= f(x)^2 -2x f(y)+y^2$ for all $x,y \in R$

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

2010 Indonesia TST, 1
Tags: algebra , functional , functional equation
Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

2020 Nordic, 4
Tags: functional equation , functional , algebra
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 $(x + 1)(y + 1)(xy + 1) \ne 0$.

2017 QEDMO 15th, 4
Tags: functional equation , functional , algebra
Find all functions $f: R \to R$ for which the image $f ([a, b])$ for all real $a \le b$ is (not necessarily closed!) interval of length $b - a$.

2014 Contests, 1b
Tags: functional equation , functional , function , algebra
Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.

2004 Thailand Mathematical Olympiad, 2
Tags: functional , functional equation , algebra
Let $f : Q \to Q$ be a function satisfying the equation $f(x + y) = f(x) + f(y) + 2547$ for all rational numbers $x, y$. If $f(2004) = 2547$, find $f(2547)$.

1989 Swedish Mathematical Competition, 2
Tags: continuous , functional equation , functional , algebra
Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.

2011 Indonesia TST, 1
Tags: algebra , functional equation , functional equation in q , functional
Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

2013 HMIC, 2
Tags: functional
Find all functions $f : R \to R$ such that, for all real numbers $x, y,$ $$(x - y)(f(x) - f(y)) = f(x - f(y))f(f(x) - y).$$

1972 Dutch Mathematical Olympiad, 2
Tags: functional , inequalities , functional equation , functional inequality , algebra
Prove that there exists exactly one function $ƒ$ which is defined for all $x \in R$, and for which holds: $\bullet$ $x \le y \Rightarrow f(x) \le f(y)$, for all $x, y \in R$, and $\bullet$ $f(f(x)) = x$, for all $x \in R$.

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 $f(p)|f(n)$.

1976 Polish MO Finals, 6
Tags: algebra , logarithm , functional
An increasing function $f : N \to R$ satisfies $$f(kl) = f(k)+ f(l)\,\,\, for \,\,\, all \,\,\, k,l \in N.$$ Show that there is a real number $p > 1$ such that $f(n) =\ log_pn$ for all $n$.

2019 Estonia Team Selection Test, 3
Tags: algebra , functional equation , functional
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.

2024 Moldova EGMO TST, 11
Tags: functional
Find all functions $f$ from the positive integers to the positive integers such that such that for all integers $x, y$ we have $$2yf(f(x^2)+x)=f(x+1)f(2xy).$$

2018 Saudi Arabia BMO TST, 4
Tags: functional , functional equation , algebra
Find all functions $f : Z \to Z$ such that $x f (2f (y) - x) + y^2 f (2x - f (y)) = \frac{(f (x))^2}{x} + f (y f (y))$ , for all $x, y \in Z$, $x \ne 0$.