Found problems: 1513
1994 Italy TST, 3
Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.
2019 Brazil Undergrad MO, 4
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any $(x, y)$ real numbers we have
$f(xf(y)+f(x))+f(y^2)=f(x)+yf(x+y)$
2015 IMO Shortlist, A5
Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.
VMEO III 2006 Shortlist, A5
Find all continuous functions $f : (0,+\infty) \to (0,+\infty)$ such that if $a, b, c$ are the lengths of the sides of any triangle then it is satisfied that $$\frac{f(a+b-c)+f(b+c-a)+f(c+a-b)}{3}=f\left(\sqrt{\frac{ab+bc+ca}{3}}\right)$$
2005 Putnam, B3
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all $x>0.$
1986 Poland - Second Round, 1
Determine all functions $ f : \mathbb{R} \to \mathbb{R} $ continuous at zero and such that for every real number $ x $ the equality holds $$ 2f(2x) = f(x) + x.$$
1993 Nordic, 1
Let $F$ be an increasing real function defined for all $x, 0 \le x \le 1$, satisfying the conditions
(i) $F (\frac{x}{3}) = \frac{F(x)}{2}$.
(ii) $F(1- x) = 1 - F(x)$.
Determine $F(\frac{173}{1993})$ and $F(\frac{1}{13})$ .
1969 IMO Longlists, 8
Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.
2012 Online Math Open Problems, 46
If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$.
[i]Author: Alex Zhu[/i]
1994 IMO Shortlist, 4
Let $ \mathbb{R}$ denote the set of all real numbers and $ \mathbb{R}^\plus{}$ the subset of all positive ones. Let $ \alpha$ and $ \beta$ be given elements in $ \mathbb{R},$ not necessarily distinct. Find all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}$ such that
\[ f(x)f(y) \equal{} y^{\alpha} f \left( \frac{x}{2} \right) \plus{} x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^\plus{}.\]
2018 Iran MO (3rd Round), 3
Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.
2011 QEDMO 9th, 7
Find all functions $f: R\to R$, such that $f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)$ for all $x, y \in R$.
Russian TST 2018, P1
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.
2021 Federal Competition For Advanced Students, P2, 4
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)
2023 IFYM, Sozopol, 2
Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that
\[
f(x) + f(y - 1) + f(f(y - f(x))) = 1
\]
for all integers $x$ and $y$.
2015 Indonesia MO, 4
Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies
\[
f(g(x)y + f(x)) = (y+2015)f(x)
\]
for every $x,y \in \mathbb{R^+} $
a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$
b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$
2018 China Team Selection Test, 4
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.
1988 Bulgaria National Olympiad, Problem 6
Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.
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] $P(1,0) =1.$
2015 Peru IMO TST, 9
Let $A$ be a finite set of functions $f: \Bbb{R}\to \Bbb{R.}$ It is known that: [list] [*] If $f, g\in A$ then $f (g (x)) \in A.$ [*] For all $f \in A$ there exists $g \in A$ such that $f (f (x) + y) = 2x + g (g (y) - x),$ for all $x, y\in \Bbb{R}.$ [/list] Let $i:\Bbb{R}\to \Bbb{R}$ be the identity function, ie, $i (x) = x$ for all $x\in \Bbb{R}.$ Prove that $i \in A.$
2022 JHMT HS, 9
There is a unique continuous function $f$ over the positive real numbers satisfying $f(4) = 1$ and
\[ 9 - (f(x))^4 = \frac{x^2}{(f(x))^2} - 2xf(x) \]
for all positive $x$. Compute the value of $\int_{0}^{140} (f(x))^3\,dx$.
1969 IMO Shortlist, 8
Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.
1998 Belarus Team Selection Test, 3
Find all continuous functions $f: R \to R$ such that $g(g(x)) = g(x)+2x$ for all real $x$.
2021 Bolivia Ibero TST, 2
Let $f: \mathbb Z^+ \to \mathbb Z$ be a function such that
[b]a)[/b] $f(p)=1$ for every prime $p$.
[b]b)[/b] $f(xy)=xf(y)+yf(x)$ for every pair of positive integers $x,y$
Find the least number $n \ge 2021$ such that $f(n)=n$
2024 239 Open Mathematical Olympiad, 1
Let $f:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ be a continuous function such that $f(0)=0$ and $$f(x)+f(f(x))+f(f(f(x)))=3x$$ for all $x>0$. Show that $f(x)=x$ for all $x>0$.