Found problems: 4776
Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > f(x)>0$ for all real numbers $x$. Show that $f(8) > 2022f(0)$.
[i]Proposed by Ethan Tan[/i]
Find the smallest positive integer $k$ for which there exists a colouring of the positive integers $\mathbb{Z}_{>0}$ with $k$ colours and a function $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ with the following two properties:
$(i)$ For all positive integers $m,n$ of the same colour, $f(m+n)=f(m)+f(n).$
$(ii)$ There are positive integers $m,n$ such that $f(m+n)\ne f(m)+f(n).$
[i]In a colouring of $\mathbb{Z}_{>0}$ with $k$ colours, every integer is coloured in exactly one of the $k$ colours. In both $(i)$ and $(ii)$ the positive integers $m,n$ are not necessarily distinct.[/i]
Find the range of the function $f(x)=\frac{\sqrt{x^2+1}}{x-1}$.
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 $x$.
If $f(x, y) = xy + 2x + y + 1$, find $f(f(2, f(3, 4)), 5)$.
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that:
$ \text{(i)}\quad f(0)=0 $
$ \text{(ii)}\quad f'(x)\neq 0,\quad\forall x\in\mathbb{R} $
$ \text{(iii)}\quad \left. f''\right|_{\mathbb{R}}\text{ exists and it's continuous} $
Demonstrate that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ defined as
$$ g(x)=\left\{\begin{matrix}\cos\frac{1}{f(x)},\quad x\neq 0\\ 0,\quad x=0\end{matrix}\right. $$
is primitivable.
Find the remainder when
\[\sum_{k=1}^{2^{16}}\binom{2k}{k}(3\cdot 2^{14}+1)^k (k-1)^{2^{16}-1}\]is divided by $2^{16}+1$. ([i]Note:[/i] It is well-known that $2^{16}+1=65537$ is prime.)
[i]Victor Wang.[/i]
Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$.
(The latter means that
$ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$
holds for every integrable function $ h(x)$.)
Find all functions $ f:\mathbb{R^{\ast }}\rightarrow \mathbb{ R^{\ast }}$ satisfying $f(\frac{f(x)}{f(y)})=\frac{1}{y}f(f(x))$ for all $x,y\in \mathbb{R^{\ast }}$
and are strictly monotone in $(0,+\infty )$
For every rational number $m>0$ we consider the function $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$. Denote by $G_m$ the graph of the function $f_m$. Let $p,q,r$ be positive rational numbers.
a) Show that if $p$ and $q$ are distinct then $G_p\cap G_q$ is non-empty.
b) Show that if $G_p\cap G_q$ is a point with integer coordinates, then $p$ and $q$ are integer numbers.
c) Show that if $p,q,r$ are consecutive natural numbers, then the area of the triangle determined by intersections of $G_p,G_q$ and $G_r$ is equal to $1$.
Range of function $f(x)=\arctan x+\frac{1}{2}\arcsin x$ is
$\text{(A)}(-\pi,\pi)\qquad\text{(B)}[-\frac{3}{4}\pi,\frac{3}{4}\pi]\qquad\text{(C)}(-\frac{3}{4}\pi,\frac{3}{4}\pi)\qquad\text{(D)}[-\frac{1}{2}\pi,\frac{1}{2}\pi]$
[b]Problem 1[/b]
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable on $\mathbb{R}$. Prove that there exists $x \in [0, 1]$ such that
$$\frac{4}{\pi} ( f(1) - f(0) ) = (1+x^2) f'(x) \ .$$
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy $yf(x)+f(y) \geq f(xy)$
Let $ R$ be a bounded domain of area $ t$ in the plane, and let $ C$ be its center of gravity. Denoting by $ T_{AB}$ the circle drawn with the diameter $ AB$, let $ K$ be a circle that contains each of the circles $ T_{AB} \;(A,B \in R)$. Is it true in general that $ K$ contains the circle of area $ 2t$ centered at $ C$?
[i]J. Szucs[/i]
In a coordinate system there are drawn the graphs of the functions $y=ax+b$ and $y=bx+a, (a\neq b)$. Their intersection is marked with red and their intersections with the $Oy$ axis are marked with blue. Everything is erased except the marked points. Using only a ruler and a compass, find the origin of the coordinate system.
The function $ f$ has the property that for each real number $ x$ in its domain, $ 1/x$ is also in its domain and
\[ f(x) \plus{} f\left(\frac {1}{x}\right) \equal{} x.
\]What is the largest set of real numbers that can be in the domain of $ f$?
$ \textbf{(A) } \{ x | x\ne 0\} \qquad \textbf{(B) } \{ x | x < 0\} \qquad \textbf{(C) }\{ x | x > 0\}\\
\textbf{(D) } \{ x | x\ne \minus{} 1 \text{ and } x\ne 0 \text{ and } x\ne 1\} \qquad \textbf{(E) } \{ \minus{} 1,1\}$
Let $n \ge 2$ be a positive integer and $\mathcal{F}$ the set of functions $f:\{1,2,\ldots,n\} \to \{1,2,\ldots,n\}$ that satisfy $f(k) \le f(k+1) \le f(k)+1,$ for all $k \in \{1,2,\ldots,n-1\}.$
a) Find the cardinal of the set $\mathcal{F}.$
b) Find the total number of fixed points of the functions in $\mathcal{F}.$
Find all function $ f: \mathbb{N}\rightarrow\mathbb{N}$ satisfy $ f(mn)\plus{}f(m\plus{}n)\equal{}f(m)f(n)\plus{}1$ for all natural number $ n$
The sequence $f_1, f_2, \cdots, f_n, \cdots $ of functions is defined for $x > 0$ recursively by
\[f_1(x)=x , \quad f_{n+1}(x) = f_n(x) \left(f_n(x) + \frac 1n \right)\]
Prove that there exists one and only one positive number $a$ such that $0 < f_n(a) < f_{n+1}(a) < 1$ for all integers $n \geq 1.$
Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a function having the following property: For any two points $A$ and $B$ in $\mathbb{R}^2$, the distance between $A$ and $B$ is the same as the distance between the points $f(A)$ and $f(B)$.
Denote the unique straight line passing through $A$ and $B$ by $l(A,B)$
(a) Suppose that $C,D$ are two fixed points in $\mathbb{R}^2$. If $X$ is a point on the line $l(C,D)$, then show that $f(X)$ is a point on the line $l(f(C),f(D))$.
(b) Consider two more point $E$ and $F$ in $\mathbb{R}^2$ and suppose that $l(E,F)$ intersects $l(C,D)$ at an angle $\alpha$. Show that $l(f(C),f(D))$ intersects $l(f(E),f(F))$ at an angle $\alpha$. What happens if the two lines $l(C,D)$ and $l(E,F)$ do not intersect? Justify your answer.
If $a$, $b$, $c>0$ and $abc=1$, $\alpha = max\{a,b,c\}$; $f,g : (0, +\infty )\to \mathbb{R}$, where $f(x)=\frac{2(x+1)^2}{x}$ and $g(x)= (x+1)\left (\frac{1}{\sqrt{x}}+1\right )^2$, then $$(a+1)(b+1)(c+1)\geq min\{ \{f(x),g(x) \}\ |\ x\in\{a,b,c\} \backslash \{ \alpha \}\} $$
Let us consider a function $f:\mathbb{N}\to\mathbb{N}$ for which $f(1)=1$, $f(2n)=f(n)$ and $f(2n+1)=f(2n)+1$. Find the number of values at which the maximum value of $f(n)$ is attained for integer $n$ satisfying $0<n<2014$.
Let $ n\geq 3 $ be an integer and let $ \mathcal{S} \subset \{1,2,\ldots, n^3\} $ be a set with $ 3n^2 $ elements. Prove that there exist nine distinct numbers $ a_1,a_2,\ldots,a_9 \in \mathcal{S} $ such that the following system has a solution in nonzero integers:
\begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 \\ a_4x + a_5 y + a_6 z &=& 0 \\ a_7x + a_8y + a_9z &=& 0. \end{eqnarray*}
[i]Marius Cavachi[/i]
Find $\lim_{n\to\infty} \left(\frac{_{3n}C_n}{_{2n}C_n}\right)^{\frac{1}{n}}$
where $_iC_j$ is a binominal coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}$.
Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions:
i) $g(2013)=g(2014) = 0,$
ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$
Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$