Found problems: 1513
2018 Estonia Team Selection Test, 4
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$
VI Soros Olympiad 1999 - 2000 (Russia), 10.3
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 $y$.
1989 Swedish Mathematical Competition, 2
Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.
2024 India IMOTC, 19
Denote by $\mathbb{S}$ the set of all proper subsets of $\mathbb{Z}_{>0}$. Find all functions $f : \mathbb{S} \mapsto \mathbb{Z}_{>0}$ that satisfy the following:\\
[color=#FFFFFF]___[/color]1. For all sets $A, B \in \mathbb{S}$ we have \[f(A \cap B) = \text{min}(f(A), f(B)).\]
[color=#FFFFFF]___[/color]2. For all positive integers $n$ we have \[\sum \limits_{X \subseteq [1, n]}
f(X) = 2^{n+1}-1.\]
(Here, by a proper subset $X$ of $\mathbb{Z}_{>0}$ we mean $X \subset \mathbb{Z}_{>0}$ with $X \ne \mathbb{Z}_{>0}$. It is allowed for $X$ to have infinite size.) \\
[i]Proposed by MV Adhitya, Kanav Talwar, Siddharth Choppara, Archit Manas[/i]
1978 IMO Shortlist, 9
Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$
I Soros Olympiad 1994-95 (Rus + Ukr), 10.6
Find all functions $f:R\to R$ such that for any real $x, y$ , $$f(x+2^y)=f(2^x)+f(y)$$
2020 Peru IMO TST, 6
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 $a+b>2019$.
1967 IMO Longlists, 50
The function $\varphi(x,y,z)$ defined for all triples $(x,y,z)$ of real numbers, is such that there are two functions $f$ and $g$ defined for all pairs of real numbers, such that
\[\varphi(x,y,z) = f(x+y,z) = g(x,y+z)\]
for all real numbers $x,y$ and $z.$ Show that there is a function $h$ of one real variable, such that
\[\varphi(x,y,z) = h(x+y+z)\]
for all real numbers $x,y$ and $z.$
2025 Kosovo National Mathematical Olympiad`, P4
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously
(i) For all $m,n \in \mathbb{N}$ we have:
$$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$
(ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.
2003 China Team Selection Test, 2
Find all functions $f,g$:$R \to R$ such that $f(x+yg(x))=g(x)+xf(y)$ for $x,y \in R$.
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 $y$, \[ f(x+y)+f(x-y)-2f(x)-2y^2=0. \]
2019 MMATHS, 4
The continuous function $f(x)$ satisfies $c^2f(x + y) = f(x)f(y)$ for all real numbers $x$ and $y,$ where $c > 0$ is a constant. If $f(1) = c$, find $f(x)$ (with proof).
VMEO I 2004, 5
Find all the functions $f:R \to R$ satisfying
$$(x + y)(f (x)-f (y)) = f (x^2) - f (y^2),\, \forall x, y \in R$$
2010 IMO Shortlist, 6
Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$
[i]Proposed by Alex Schreiber, Germany[/i]
2017 Abels Math Contest (Norwegian MO) Final, 1b
Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.
2007 Thailand Mathematical Olympiad, 9
Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?
2017 Dutch IMO TST, 4
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$
for all $x, y \in \mathbb{R}$.
2010 Contests, 3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.
[i]George Xing.[/i]
2023 Balkan MO Shortlist, A3
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$.
2023 OMpD, 1
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that, for all real numbers $x$ and $y$, $$f(x)(x+f(f(y))) = f(x^2)+xf(y)$$
2003 IMO Shortlist, 2
Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that
(i) $f(0) = 0, f(1) = 1;$
(ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$.
[i]Proposed by A. Di Pisquale & D. Matthews, Australia[/i]
2006 Thailand Mathematical Olympiad, 6
A function $f : R \to R$ has $f(1) < 0$, and satisfy the functional equation $$f(\cos (x + y)) = (\cos x)f(\cos y) + 2f(\sin x)f(\sin y)$$ for all reals $x, y$. Compute $f \left(\frac{2006}{2549 }\right)$
2021 Albanians Cup in Mathematics, 4
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$ satisfies,
$$2+f(x)f(y)\leq xy+2f(x+y+1).$$
2010 Contests, 2
Find all polynomials $p(x)$ with real coe ffcients such that
\[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\]
for all $a, b, c\in\mathbb{R}$.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]
2014 NIMO Problems, 6
Let $P(x)$ be a polynomial with real coefficients such that $P(12)=20$ and \[ (x-1) \cdot P(16x)= (8x-1) \cdot P(8x) \] holds for all real numbers $x$. Compute the remainder when $P(2014)$ is divided by $1000$.
[i]Proposed by Alex Gu[/i]