Found problems: 66
2024 Iran MO (3rd Round), 1
Suppose that $T\in \mathbb N$ is given. Find all functions $f:\mathbb Z \to \mathbb C$ such that, for all $m\in \mathbb Z$ we have $f(m+T)=f(m)$ and:
$$\forall a,b,c \in \mathbb Z: f(a)\overline{f(a+b)f(a+c)}f(a+b+c)=1.$$
Where $\overline{a}$ is the complex conjugate of $a$.
2005 VJIMC, Problem 2
Let $f:A^3\to A$ where $A$ is a nonempty set and $f$ satisfies:
(a) for all $x,y\in A$, $f(x,y,y)=x=f(y,y,x)$ and
(b) for all $x_1,x_2,x_3,y_1,y_2,y_3,z_1,z_2,z_3\in A$,
$$f(f(x_1,x_2,x_3),f(y_1,y_2,y_3),f(z_1,z_2,z_3))=f(f(x_1,y_1,z_1),f(x_2,y_2,z_2),f(x_3,y_3,z_3)).$$
Prove that for an arbitrary fixed $a\in A$, the operation $x+y=f(x,a,y)$ is an Abelian group addition.
2017-IMOC, N3
Find all functions $f:\mathbb N\to\mathbb N_0$ such that for all $m,n\in\mathbb N$,
\begin{align*}
f(mn)&=f(m)f(n)\\
f(m+n)&=\min(f(m),f(n))\qquad\text{if }f(m)\ne f(n)\end{align*}
2018 Ramnicean Hope, 1
Let be two nonzero real numbers $ a,b $ such that $ |a|\neq |b| $ and let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function satisfying the functional relation
$$ af(x)+bf(-x)=(x^3+x)^5+\sin^5 x . $$
Calculate $ \int_{-2019}^{2019}f(x)dx . $
[i]Constantin Rusu[/i]
2018-IMOC, A4
Find all functions $f:\mathbb R\to\mathbb R$ such that
$$f\left(x^2+f(y)\right)-y=(f(x+y)-y)^2$$holds for all $x,y\in\mathbb R$.
2024 Rioplatense Mathematical Olympiad, 5
Let $S = \{2, 3, 4, \dots\}$ be the set of positive integers greater than 1. Find all functions $f : S \to S$ that satisfy
\[
\text{gcd}(a, f(b)) \cdot \text{lcm}(f(a), b) = f(ab)
\]
for all pairs of integers $a, b \in S$.
Clarification: $\text{gcd}(a,b)$ is the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ is the least common multiple of $a$ and $b$.
2017-IMOC, A5
Find all functions $f:\mathbb Z\to\mathbb Z$ such that
$$f(mf(n+1))=f(m+1)f(n)+f(f(n))+1$$for all integer pairs $(m,n)$.
2018-IMOC, N2
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$
for all $x,y\in\mathbb N$.
2014 BMT Spring, 6
Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation
$$4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1.$$
2018-IMOC, N1
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$x+f^{f(x)}(y)\mid2(x+y)$$for all $x,y\in\mathbb N$.
2010 IMO Shortlist, 1
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
1999 Mongolian Mathematical Olympiad, Problem 2
Find all functions $f:\mathbb R\to\mathbb R$ such that
(i) $f(0)=1$;
(ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$;
(iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.
2000 Moldova National Olympiad, Problem 5
Prove that there is no polynomial $P(x)$ with real coefficients that satisfies
$$P'(x)P''(x)>P(x)P'''(x)\qquad\text{for all }x\in\mathbb R.$$Is this statement true for all of the thrice differentiable real functions?
1973 Bulgaria National Olympiad, Problem 4
Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy
$$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$
for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$.
[i]L. Davidov[/i]
2020 Hong Kong TST, 4
Find all real-valued functions $f$ defined on the set of real numbers such that $$f(f(x)+y)+f(x+f(y))=2f(xf(y))$$ for any real numbers $x$ and $y$.
1998 Croatia National Olympiad, Problem 3
Let $A=\{1,2,\ldots,2n\}$ and let the function $g:A\to A$ be defined by $g(k)=2n-k+1$. Does there exist a function $f:A\to A$ such that $f(k)\ne g(k)$ and $f(f(f(k)))=g(k)$ for all $k\in A$, if (a) $n=999$; (b) $n=1000$?
2019-IMOC, A4
Find all functions $f:\mathbb N\to\mathbb N$ so that
$$f^{2f(b)}(2a)=f(f(a+b))+a+b$$
holds for all positive integers $a,b$.
2020 Jozsef Wildt International Math Competition, W44
We consider a function $f:\mathbb R\to\mathbb R$ such that
$$f(x+y)+f(xy-1)=f(x)f(y)+f(x)+f(y)+1$$
for each $x,y\in\mathbb R$.
i) Calculate $f(0)$ and $f(-1)$.
ii) Prove that $f$ is an even function.
iii) Give an example of such a function.
iv) Find all monotone functions with the above property.
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
2010 Contests, 1
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
2023 APMO, 4
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
2023 Brazil Team Selection Test, 4
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
2019 APMO, 5
Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that
\[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \]
for all real numbers $x$ and $y$.
2000 Brazil Team Selection Test, Problem 2
Find all functions $f:\mathbb R\to\mathbb R$ such that
(i) $f(0)=1$;
(ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$;
(iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.
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.
2008 VJIMC, Problem 1
Find all functions $f:\mathbb Z\to\mathbb Z$ such that
$$19f(x)-17f(f(x))=2x$$for all $x\in\mathbb Z$.