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

2003 Switzerland Team Selection Test, 10
Tags: functional equation , functional , algebra
Find all strictly monotonous functions $f : N \to N$ that satisfy $f(f(n)) = 3n$ for all $n \in N$.

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 $xyz = 1$.

2008 Indonesia TST, 2
Tags: algebra , functional equation , functional
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 $x, y$.

2021 Ukraine National Mathematical Olympiad, 4
Tags: functional equation , functional , algebra
Find all the following functions $f:R\to R$ , which for arbitrary valid $x,y$ holds equality: $$f(xf(x+y))+f((x+y)f(y))=(x+y)^2$$ (Vadym Koval)

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) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$

2004 Thailand Mathematical Olympiad, 8
Tags: algebra , functional , functional equation
Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.

2015 Saudi Arabia BMO TST, 1
Tags: algebra , functional equation , functional , function , increasing
Find all strictly increasing functions $f : Z \to R$ such that for any $m, n \in Z$ there exists a $k \in Z$ such that $f(k) = f(m) - f(n)$. Nguyễn Duy Thái Sơn

2016 Thailand Mathematical Olympiad, 9
Tags: functional equation , functional , algebra
A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.

1994 Swedish Mathematical Competition, 6
Tags: number theory , functional , functional equation
Let $N$ be the set of non-negative integers. The function $f:N\to N$ satisfies $f(a+b) = f(f(a)+b)$ for all $a, b$ and $f(a+b) = f(a)+f(b)$ for $a+b < 10$. Also $f(10) = 1$. How many three digit numbers $n$ satisfy $f(n) = f(N)$, where $N$ is the "tower" $2, 3, 4, 5$, in other words, it is $2^a$, where $a = 3^b$, where $b = 4^5$?

2020 Dutch IMO TST, 3
Tags: functional equation , functional equation in z , functional , algebra
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

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: $$f(2^x+2y) =2^yf(f(x))f(y).$$

1994 Austrian-Polish Competition, 1
Tags: functional , functional inequality , functional equation , algebra
A function $f: R \to R$ satisfies the conditions: $f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$. Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.

2018 Swedish Mathematical Competition, 2
Tags: algebra , functional equation , functional
Find all functions $f: R \to R$ that satisfy $f (x) + 2f (\sqrt[3]{1-x^3}) = x^3$ for all real $x$. (Here $\sqrt[3]{x}$ is defined all over $R$.)

1968 German National Olympiad, 3
Tags: algebra , functional , functional equation
Specify all functions $y = f(x)$, each in the largest possible domain (within the range of real numbers) of the equation $$a \cdot f(x^n) + f(-x^n) = bx$$ suffice, where $b$ is any real number, $n$ is any odd natural number and $a$ is a real number with $|a| \ne 1$.

1993 Tournament Of Towns, (377) 5
Tags: function , functional , functional equation , algebra
Does there exist a piecewise linear function $f$ defined on the segment [$-1,1]$ (including the ends) such that $f(f(x)) = -x$ for all x? (A function is called piecewise linear if its graph is the union of a finite set of points and intervals; it may be discontinuous).

1999 Austrian-Polish Competition, 3
Tags: system of equations , functional equation , functional , algebra
Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$ $$\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}$$

2021 Federal Competition For Advanced Students, P2, 4
Tags: functional equation , functional , algebra
Let $a$ be a real number. Determine all functions $f: R \to R$ with $f (f (x) + y) = f (x^2 - y) + af (x) y$ for all $x, y \in R$. (Walther Janous)

2015 Ukraine Team Selection Test, 8
Tags: functional equation , functional , algebra
Find all functions $f: R \to R$ such that $f(x)f(yf(x)-1)=x^2f(y)-f(x)$ for all real $x ,y$

2008 Indonesia TST, 2
Tags: algebra , functional equation , functional
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 $x, y$.

2020 Estonia Team Selection Test, 3
Tags: algebra , functional , functional equation
Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$

1990 Greece National Olympiad, 3
Tags: function , algebra , inequalities , functional
Find all functions $f: \mathbb{R}\to\mathbb{R}$ that satisfy $y^2f(x)(f(x)-2x)\le (1-xy)(1+xy) $ for any $x,y \in\mathbb{R}$.

2020 Greece Team Selection Test, 1
Tags: algebra , functional , functional equation
Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

2020 Greece Team Selection Test, 1
Tags: algebra , functional , functional equation
Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

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 Switzerland - Final Round, 4
Tags: algebra , functional , functional equation
Find all functions $f : R_{>0} \to R_{>0}$ with the following property: $$f \left( \frac{x}{y + 1}\right) = 1 - xf(x + y)$$ for all $x > y > 0$ .