Found problems: 1513
2004 IMO Shortlist, 6
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[
f(x^2+y^2+2f(xy)) = (f(x+y))^2.
\] for all $x,y \in \mathbb{R}$.
2020 Grand Duchy of Lithuania, 1
Find all functions $f: R \to R$, such that equality $f (xf (y) - yf (x)) = f (xy) - xy$ holds for all $x, y \in R$.
2008 Thailand Mathematical Olympiad, 6
Let $f : R^+ \to R^+$ satisfy $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$ with $x^2y^3 > 2008.$
Prove that $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$.
1995 Grosman Memorial Mathematical Olympiad, 6
(a) Prove that there is a unique function $f : Q \to Q$ satisfying:
(i) $f(q)= 1 + f\left(\frac{q}{1-2q}\right)$ for $0<q< \frac12$
(ii) $f(q)= 1 + f(q-1)$ for $1<q\le 2$
(iii) $f(q)f\left(\frac{1}{q}\right)=1$ for all $q\in Q^+$
(b) For this function $f$ , find all $r\in Q^+$ such that $f(r) = r$
2004 VJIMC, Problem 2
Find all functions $f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}$ such that
$1$. $f(x,0)=f(0,x)=x$ for all $x\in\mathbb R_{\ge0}$,
$2$. $f(f(x,y),z)=f(x,f(y,z))$ for all $x,y,z\in\mathbb R_{\ge0}$ and
$3$. there exists a real $k$ such that $f(x+y,x+z)=kx+f(y,z)$ for all $x,y,z\in\mathbb R_{\ge0}$.
2005 IMO Shortlist, 4
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$.
[i]Proposed by B.J. Venkatachala, India[/i]
2006 Italy TST, 3
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$,
\[f(m - n + f(n)) = f(m) + f(n).\]
1981 IMO Shortlist, 7
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.
1992 IMO Longlists, 51
Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose
\[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\]
Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$
2016 Abels Math Contest (Norwegian MO) Final, 4
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x) f(y) = |x - y| \cdot f \left( \frac{xy + 1}{x - y} \right) \]
Holds for all $x \not= y \in \mathbb{R}$
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 $i \in A.$
2011 ELMO Problems, 4
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
1986 Polish MO Finals, 4
Find all $n$ such that there is a real polynomial $f(x)$ of degree $n$ such that $f(x) \ge f'(x)$ for all real $x$.
1988 IMO Longlists, 49
Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$
2016 USA Team Selection Test, 2
Let $n \ge 4$ be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition $[n] = A \cup B \cup C$ into disjoint sets, \[ \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c) = |A| |B| |C|. \]
2024 USAJMO, 5
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy
\[
f(x^2-y)+2yf(x)=f(f(x))+f(y)
\]
for all $x,y\in\mathbb{R}$.
[i]Proposed by Carl Schildkraut[/i]
2012 Hanoi Open Mathematics Competitions, 5
Let $f(x)$ be a function such that $f(x)+2f\left(\frac{x+2010}{x-1}\right)=4020 - x$ for all $x \ne 1$.
Then the value of $f(2012)$ is
(A) $2010$, (B) $2011$, (C) $2012$, (D) $2014$, (E) None of the above.
2021 Korea Junior Math Olympiad, 5
Determine all functions $f \colon \mathbb{R} \to \mathbb{R}$ satisfying
$$f(f(x+y)-f(x-y))=y^2f(x)$$
for all $x, y \in \mathbb{R}$.
1995 Austrian-Polish Competition, 4
Determine all polynomials $P(x)$ with real coefficients such that
$P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$.
I Soros Olympiad 1994-95 (Rus + Ukr), 11.5
Is there a function $f(x)$ defined for all $x$ and such that for some $a$ and all $x$ holds the equality
$$f(x) + f(2x^2 - 1) = 2x + a?$$
Russian TST 2019, P1
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
2013 Albania Team Selection Test, 3
Solve the function $f: \Re \to \Re$:
\[ f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))\]
2010 Albania Team Selection Test, 2
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$,
$(1+f(x)f(y))f(x+y)=f(x)+f(y)$.
1997 Rioplatense Mathematical Olympiad, Level 3, 6
Let $N$ be the set of positive integers.
Determine if there is a function $f: N\to N$ such that $f(f(n))=2n$, for all $n$ belongs to $N$.
2023 Indonesia TST, A
Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied
\[f(x+y) + f(x)f(y) = f(xy) + 1 \]
$\forall x, y \in \mathbb{R}$