Found problems: 4776
2000 JBMO ShortLists, 11
Prove that for any integer $n$ one can find integers $a$ and $b$ such that
\[n=\left[ a\sqrt{2}\right]+\left[ b\sqrt{3}\right] \]
2010 Iran Team Selection Test, 2
Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$
\[f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).\]
2012 Kazakhstan National Olympiad, 1
Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.
2006 China Team Selection Test, 3
$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition:
(1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$
(2) $d \mid (x_1+x_2+ \cdots x_n)$
Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.
2005 Italy TST, 3
The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$.
$(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$.
$(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.
2017 Iran Team Selection Test, 3
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$
$$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$
$$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$
[i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]
The Golden Digits 2024, P1
Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ with the following properties:
1) For every natural number $n\geq 3$, $\gcd(f(n),n)\neq 1$.
2) For every natural number $n\geq 3$, there exists $i_n\in\mathbb{Z}_{>0}$, $1\leq i_n\leq n-1$, such that $f(n)=f(i_n)+f(n-i_n)$.
[i]Proposed by Pavel Ciurea[/i]
1997 AMC 12/AHSME, 21
For any positive integer $ n$, let \[f(n) \equal{} \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\
0, & \text{otherwise.} \end{cases}\] What is $ \sum_{n \equal{} 1}^{1997}{f(n)}$?
$ \textbf{(A)}\ \log_8{2047}\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ \frac {55}{3}\qquad \textbf{(D)}\ \frac {58}{3}\qquad \textbf{(E)}\ 585$
2018 Ukraine Team Selection Test, 12
Let $n$ be a positive integer and $a_1,a_2,\dots,a_n$ be integers. Function $f: \mathbb{Z} \rightarrow \mathbb{R}$ is such that for all integers $k$ and $l$, $l \neq 0$, $$\sum_{i=1}^n f(k+a_il)=0.$$ Prove that $f \equiv 0$.
2023 Indonesia TST, A
Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfied
\[f(x+y) + f(x)f(y) = f(xy) + 1 \]
$\forall x, y \in \mathbb{R}$
2014 District Olympiad, 1
For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n\rightarrow\infty}{\lim}\frac{1}{n}\int_{0}^{n}f_{n}(x)\mathrm{d}x.$
1981 Miklós Schweitzer, 8
Let $ W$ be a dense, open subset of the real line $ \mathbb{R}$. Show that the following two statements are equivalent:
(1) Every function $ f : \mathbb{R} \rightarrow \mathbb{R}$ continuous at all points of $ \mathbb{R} \setminus W$ and nondecreasing on every open interval contained in $ W$ is nondecreasing on the whole $ \mathbb{R}$.
(2) $ \mathbb{R} \setminus W$ is countable.
[i]E. Gesztelyi[/i]
2008 China Team Selection Test, 5
For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where
\[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]
2021 Indonesia TST, A
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x + y) + y \le f(f(f(x)))\]
holds for all $x, y \in \mathbb{R}$.
2019 India PRMO, 2
Ket $f(x) = x^{2} +ax + b$. If for all nonzero real $x$
$$f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right)$$
and the roots of $f(x) = 0$ are integers, what is the value of $a^{2}+b^{2}$?
1975 Spain Mathematical Olympiad, 2
Study the real function $f(x) = \left(1 +\frac{1}{x}\right)^x$ defined for $ x \in R - \{-1, 0\}$ . Graphic representation.
1958 Miklós Schweitzer, 8
[b]8.[/b] Let the function $f(x)$ be periodic with the period $1$, non-negative, concave in the interval $(0,1)$ and continuous at the point $0$. Prove that $f(nx)\leq nf(x)$ for every real $x$ and positive integer $n$. [b](R. 6)[/b]
1940 Putnam, A3
Let $a$ be a real number. Find all real-valued functions $f$ such that
$$\int f(x)^{a} dx=\left( \int f(x) dx \right)^{a}$$
when constants of integration are suitably chosen.
1983 National High School Mathematics League, 5
$f(x)=ax^2-c$. If$-4\leq f(1)\leq -1,-z\leq f(2)\leq 5$, then
$\text{(A)}7\leq f(3)\leq26\qquad\text{(B)}-4\leq f(3)\leq15\qquad\text{(C)}-1\leq f(3)\leq23\qquad\text{(D)}-\frac{28}{3}\leq f(3)\leq\frac{35}{3}$
2018 Bundeswettbewerb Mathematik, 2
Consider all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying $f(1-f(x))=x$ for all $x \in \mathbb{R}$.
a) By giving a concrete example, show that such a function exists.
b) For each such function define the sum
\[S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018).\]
Determine all possible values of $S_f$.
2021 JHMT HS, 4
There is a unique differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ satisfying $f(x) + (f(x))^3 = x + x^7$ for all real $x.$ The derivative of $f(x)$ at $x = 2$ can be expressed as a common fraction $a/b.$ Compute $a + b.$
1993 Miklós Schweitzer, 4
Let f be a ternary operation on a set of at least four elements for which
(1) $f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x$
(2) $f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}$
for pairwise distinct x,y,z.
Prove that f is a nontrivial composition of g such that g is not a composition of f.
(The n-variable operation g is trivial if $g(x_1, ..., x_n) \equiv x_i$ for some i ($1 \leq i \leq n$) )
2014 ELMO Shortlist, 5
Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]
2011 Today's Calculation Of Integral, 721
For constant $a$, find the differentiable function $f(x)$ satisfying $\int_0^x (e^{-x}-ae^{-t})f(t)dt=0$.
2010 Indonesia TST, 1
Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number?
[i]Nanang Susyanto, Jogjakarta[/i]