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

2005 Chile National Olympiad, 5
Tags: functional , number theory
Compute $g(2005)$ where $g$ is a function defined on the natural numbers that has the following properties: i) $g(1) = 0$ ii) $g(nm) = g(n) + g(m) + g(n)g(m)$ for any pair of integers $n, m$. iii) $g(n^2 + 1) = (g(n) + 1)^2$ for every integer $n$.

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

2021 Francophone Mathematical Olympiad, 4
Tags: least common multiple , functional , functional equation , greatest common divisor , number theory
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 $b$.

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 $x, y \in Z$, $x \ne 0$.

VMEO III 2006 Shortlist, A6
Tags: functional , algebra , functional equation
The symbol $N_m$ denotes the set of all integers not less than the given integer $m$. Find all functions $f: N_m \to N_m$ such that $f(x^2+f(y))=y^2+f(x)$ for all $x,y \in N_m$.

2017 Thailand Mathematical Olympiad, 3
Tags: functional equation , functional , algebra
Determine all functions $f : R \to R$ satisfying $f(f(x) - y) \le xf(x) + f(y)$ for all real numbers $x, y$.

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

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

1990 Romania Team Selection Test, 3
Tags: functional equation , functional , algebra , polynomial
Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

1968 German National Olympiad, 3
Tags: algebra , functional equation , functional
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 $|a| \ne 1$.

2017 Singapore Senior Math Olympiad, 4
Tags: inequalities , functional , functional inequality , algebra
Find all functions $f : Z^+ \to Z^+$ such that $f(k + 1) >f(f(k))$ for $k > 1$, where $Z^+$ is the set of positive integers.

1998 Belarusian National Olympiad, 8
Tags: algebra , functional , functional equation
a) Prove that for no real a such that $0 \le a <1$ there exists a function defined on the set of all positive numbers and taking values in the same set, satisfying for all positive $x$ the equality $$f\left(f(x)+\frac{1}{f(x)}\right)=x+a \,\,\,\,\,\,\, (*) $$ b) Prove that for any $a>1$ there are infinitely many functions defined on the set of all positive numbers, with values in the same set, satisfying ($*$) for all positive x.

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

2013 QEDMO 13th or 12th, 4
Tags: functional , algebra , periodic , functional equation
Let $a> 0$ and $f: R\to R$ a function such that $f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1$ for all $x\in R$ . Show that $f$ is periodic, that is, that there is some $b> 0$ for which $f (x) = f (x + b)$ for every $x \in R$ holds. Find the smallest such $b$, which works for all these functions .

2013 Chile National Olympiad, 4
Tags: functional equation , functional , 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 $f(m) = 2m$.

2000 Belarus Team Selection Test, 4.1
Tags: functional , algebra , function , functional equation
Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$

2020 Swedish Mathematical Competition, 3
Tags: algebra , bounded , functional , isl , functional equation
Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.

1993 Tournament Of Towns, (377) 5
Tags: algebra , functional equation , functional , function
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).

1965 Dutch Mathematical Olympiad, 5
Tags: functional , functional equation , algebra
The function ƒ. which is defined for all real numbers satisfies: $$f(x+y)+f(x-y)=2f(x)+2f(y)$$ Prove that $f(0) = 0$, $f(-x) = f(x)$, $f(2x) = 4 f (x)$, $$f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).$$

2020 Greece Team Selection Test, 1
Tags: functional , algebra , 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)

2014 Belarus Team Selection Test, 1
Tags: functional equation , functional , algebra
Find all functions$ f : R_+ \to R_+$ such that $f(f(x)+y)=x+f(y)$ , for all $x, y \in R_+$ (Folklore) [hide=PS]Using search terms [color=#f00]+ ''f(x+f(y))'' + ''f(x)+y[/color]'' I found the same problem [url=https://artofproblemsolving.com/community/c6h1122140p5167983]in Q[/url], [url=https://artofproblemsolving.com/community/c6h1597644p9926878]continuous in R[/url], [url=https://artofproblemsolving.com/community/c6h1065586p4628238]strictly monotone in R[/url] , [url=https://artofproblemsolving.com/community/c6h583742p3451211 ]without extra conditions in R[/url] [/hide]

2011 Cuba MO, 5
Tags: functional , algebra , functional equation
Determine all functions $f : R \to R$ such that $$f(x)f(y) = 2f(x + y) + 9xy \ \ \forall x, y \in R.$$

I Soros Olympiad 1994-95 (Rus + Ukr), 11.1
Tags: functional , algebra
Let the function $f:R \to R$ satisfies the following conditions: 1) for all $x, y\in R$, $ f(x +y) = f(x) +f(y)$ 2)$ f(1)=1$ 3) for all $x \ne 0$ , $ f(1/x) =\frac{f(x)}{x^2}$ Prove that for all $x \in R$, $f(x) = x$.

1999 Switzerland Team Selection Test, 3
Tags: functional , functional equation , 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 $x \ne 0$.

2017 QEDMO 15th, 8
Tags: functional equation , functional , algebra
For a function $f: R\to R $ , $ f (2017)> 0$ as well as $f (x^2 + yf (z)) = xf (x) + zf (y)$ for all $x,y,z \in R$ is known. What is the value of $f (-42)$?