Found problems: 66
2018-IMOC, A1
Find all functions $f:\mathbb Q\to\mathbb Q$ such that for all $x,y,z,w\in\mathbb Q$,
$$f(f(xyzw)+x+y)+f(z)+f(w)=f(f(xyzw)+z+w)+f(x)+f(y).$$
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]
2014 BMT Spring, 6
Find $f(2)$ given that $f$ is a real-valued function that satisfies the equation
$$4f(x)+\left(\frac23\right)(x^2+2)f\left(x-\frac2x\right)=x^3+1.$$
2019 Romanian Master of Mathematics, 5
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying
\[f(x + yf(x)) + f(xy) = f(x) + f(2019y),\]
for all 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) $n=1000$?
2018-IMOC, A3
Find all functions $f:\mathbb R\to\mathbb R$ such that for reals $x,y$,
$$f(xf(y)+y)=yf(x)+f(y).$$
2025 Nordic, 1
Let $n$ be a positive integer greater than $2$. Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying:
$(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$
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 $a,b$.
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]
2000 Mongolian Mathematical Olympiad, Problem 4
Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions:
(i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$;
(ii) $f(f(f(0)))=0$.
Prove that $f(0)=0$.
2022 Indonesia TST, A
Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying
\[ f(a^2) - f(b^2) \leq (f(a)+b)(a-f(b)) \] for all $a,b \in \mathbb{R}$.
2010 IMO, 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]
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?
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 $x,y,z\in\mathbb R_{\ge0}$.
1997 Croatia National Olympiad, Problem 3
Function $f$ is defined on the positive integers by $f(1)=1$, $f(2)=2$ and
$$f(n+2)=f(n+2-f(n+1))+f(n+1-f(n))\enspace\text{for }n\ge1.$$
(a) Prove that $f(n+1)-f(n)\in\{0,1\}$ for each $n\ge1$.
(b) Show that if $f(n)$ is odd then $f(n+1)=f(n)+1$.
(c) For each positive integer $k$ find all $n$ for which $f(n)=2^{k-1}+1$.
2014 BMT Spring, 8
Suppose an integer-valued function $f$ satisfies
$$\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1}$$
for all non-negative integers $n$. Determine $\sum_{n=0}^\infty\frac{f(n)}{2^n}$.
1998 Slovenia National Olympiad, Problem 2
Find all polynomials $p$ with real coefficients such that for all real $x$
$$(x-8)p(2x)=8(x-1)p(x).$$
2005 Federal Math Competition of S&M, Problem 3
Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and
$$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.
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 $b$.
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.
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 $x$.
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 1
Suppose that a function $f:\mathbb R\to\mathbb R$ is such that for any real $h$ there exist at most $19990509$ different values of $x$ for which $f(x)\ne f(x+h)$. Prove that there is a set of at most $9995256$ real numbers such that $f$ is constant outside of this set.
2017-IMOC, N5
Find all functions $f:\mathbb N\to\mathbb N$ such that
$$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.