Found problems: 1513
2011 Ukraine Team Selection Test, 5
Denote by $\mathbb{Q}^+$ the set of all positive rational numbers. Determine all functions $f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+$ which satisfy the following equation for all $x, y \in \mathbb{Q}^+:$ \[f\left( f(x)^2y \right) = x^3 f(xy).\]
[i]Proposed by Thomas Huber, Switzerland[/i]
2015 Iran MO (3rd round), 5
Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$:
$p(5x)^2-3=p(5x^2+1)$ such that:
$a) p(0)\neq 0$
$b) p(0)=0$
2022 Taiwan TST Round 3, 4
Let $\mathcal{X}$ be the collection of all non-empty subsets (not necessarily finite) of the positive integer set $\mathbb{N}$. Determine all functions $f: \mathcal{X} \to \mathbb{R}^+$ satisfying the following properties:
(i) For all $S$, $T \in \mathcal{X}$ with $S\subseteq T$, there holds $f(T) \le f(S)$.
(ii) For all $S$, $T \in \mathcal{X}$, there hold
\[f(S) + f(T) \le f(S + T),\quad f(S)f(T) = f(S\cdot T), \]
where $S + T = \{s + t\mid s\in S, t\in T\}$ and $S \cdot T = \{s\cdot t\mid s\in S, t\in T\}$.
[i]Proposed by Li4, Untro368, and Ming Hsiao.[/i]
1975 IMO, 6
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
2016 Bosnia and Herzegovina Team Selection Test, 6
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
2001 Czech And Slovak Olympiad IIIA, 1
Determine all polynomials $P$ such that for every real number $x$, $P(x)^2 +P(-x) = P(x^2)+P(x)$
2024 Pan-American Girls’ Mathematical Olympiad, 5
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$
for all real numbers $x, y$.
2012 Poland - Second Round, 1
$f,g:\mathbb{R}\rightarrow\mathbb{R}$ find all $f,g$ satisfying $\forall x,y\in \mathbb{R}$:
\[g(f(x)-y)=f(g(y))+x.\]
2010 IMAC Arhimede, 2
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$
2018 Poland - Second Round, 1
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfy conditions:
$f(x) + f(y) \ge xy$ for all real $x, y$ and
for each real $x$ exists real $y$, such that $f(x) + f(y) = xy$.
Russian TST 2020, P2
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
2012 Brazil National Olympiad, 6
Find all surjective functions $f\colon (0,+\infty) \to (0,+\infty)$ such that $2x f(f(x)) = f(x)(x+f(f(x)))$ for all $x>0$.
2022 Swedish Mathematical Competition, 2
Find all functions $f : R \to R$ such that $$f(x + zf(y)) = f(x) + zf(y), $$ for all $x, y, z \in R$.
2014 Contests, 2
Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions:
(i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and
(ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?
[i]Proposed by Igor I. Voronovich, Belarus[/i]
2010 Contests, 1
Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.
2005 Korea National Olympiad, 4
Find all $f: \mathbb R \to\mathbb R$ such that for all real numbers $x$, $f(x) \geq 0$ and for all real numbers $x$ and $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]
2011 Postal Coaching, 3
Suppose $f : \mathbb{R} \longrightarrow \mathbb{R}$ be a function such that
\[2f (f (x)) = (x^2 - x)f (x) + 4 - 2x\]
for all real $x$. Find $f (2)$ and all possible values of $f (1)$. For each value of $f (1)$, construct a function achieving it and satisfying the given equation.
2016 India IMO Training Camp, 2
Find all functions $f:\mathbb R\to\mathbb R$ such that $$f\left( x^2+xf(y)\right)=xf(x+y)$$ for all reals $x,y$.
2018 Pan-African Shortlist, A1
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $(f(x + y))^2 = f(x^2) + f(y^2)$ for all $x, y \in \mathbb{Z}$.
1998 IMO, 6
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$,
\[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]
2004 Germany Team Selection Test, 2
Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties:
(a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$.
(b) We have $f\left(2\right) = 0$.
(c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$.
[b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.
2008 Bulgarian Autumn Math Competition, Problem 12.3
Find all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that
\[(f(x)f(y)-1)f(x+y)=2f(x)f(y)-f(x)-f(y)\quad \forall x,y\in \mathbb{R}\]
2018 Estonia Team Selection Test, 4
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$
2012 IMO Shortlist, A5
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that satisfy the conditions
\[f(1+xy)-f(x+y)=f(x)f(y) \quad \text{for all } x,y \in \mathbb{R},\]
and $f(-1) \neq 0$.
2011 International Zhautykov Olympiad, 2
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy the equality,
\[f(x+f(y))=f(x-f(y))+4xf(y)\]
for any $x,y\in\mathbb{R}$.