Found problems: 4776
A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors).
Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.)
[hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]
Let $ f:\mathbb{R}\longrightarrow (0,\infty ) $ be a continuous and periodic function having a period of $ 2, $ and such that the integral $ \int_0^2 \frac{f(1+x)}{f(x)} dx $ exists. Show that
$$ \int_0^2 \frac{f(1+x)}{f(x)} dx\ge 2, $$
with equality if and only if $ 1 $ is also a period of $ f. $
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that
\[ f'\left(\frac ax\right)=\frac x{f(x)} \]
for all $x>0.$
If $a>0,a\neq1$, $F(x)$ is an odd function. $G(x)=F(x)\cdot(\frac{1}{a^x-1}+\frac{1}{2})$, then $G(x)$ is
$\text{(A)}$ odd function
$\text{(B)}$ even function
$\text{(C)}$ not odd or even function
$\text{(D)}$ not sure
Prove that all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying, for all real $x$,
\[ f(x)=f(2x)=f(1-x)\]
are periodic.
Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds:
$\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$,
where $T$ is the period of $f(x)$.
Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation
\[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]
Define a function $g: \mathbb{N} \mapsto \mathbb{N}$ by the following rule:
(a) $g$ is nondecrasing
(b) for each $n$, $g(n)$ i sthe number of times $n$ appears in the range of $g$,
Prove that $g(1) = 1$ and $g(n+1) = 1 + g( n +1 - g(g(n)))$ for all $n \in \mathbb{N}$
Prove that
\[\arctan \frac 12 +\arctan \frac 13 = \frac{\pi}{4}.\]
Let $ABCD$ be a parallelogram that is not rhombus. We draw the symmetrical half-line of $(DC$ with respect to line $BD$. Similarly we draw the symmetrical half- line of $(AB$ with respect to $AC$. These half- lines intersect each other in $P$. If $\frac{AP}{DP}= q$ find the value of $\frac{AC}{BD}$ in function of $q$.
Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying
\[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\]
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]
Let $n$ is positive integer, $D_n$ is a set of all positive divisor of $n$ and $f(n)=\sum_{d\in D_n}{\frac{1}{1+d}}$
Prove that for all positive integer $m$, $\sum_{i=1}^{m}{f(i)} <m$
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that
[list]
[*] $f(x \cdot y) = f(x) + f(y)$
[*] $f(30) = 0$
[*] $f(x)=0$ always when the units digit of $x$ is $7$
[/list]
Let $n$ be a positive integer, let $f_1(x),\ldots,f_n(x)$ be $n$ bounded real functions, and let $a_1,\ldots,a_n$ be $n$ distinct reals.
Show that there exists a real number $x$ such that $\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1$.
In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.
Let $\mathbb{F}_p$ denote the field of integers modulo a prime $p,$ and let $n$ be a positive integer. Let $v$ be a fixed vector in $\mathbb{F}_p^n,$ let $M$ be an $n\times n$ matrix with entries in $\mathbb{F}_p,$ and define $G:\mathbb{F}_p^n\to \mathbb{F}_p^n$ by $G(x)=v+Mx.$ Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x)=G(x)$ and $G^{(k+1)}(x)=G(G^{(k)}(x)).$ Determine all pairs $p,n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0),$ $k=1,2,\dots,p^n$ are distinct.
Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.
Let$f:\mathbb{R}\to \mathbb{R}$be a monotone function.
a) Prove that$f$ have side limits in each point ${{x}_{0}}\in \mathbb{R}$.
b) We define the function $g:\mathbb{R}\to \mathbb{R}$, $g\left( x \right)=\underset{t\nearrow x}{\mathop{\lim }}\,f\left( t \right)$( $g\left( x \right)$ with limit at at left in $x$). Prove that if the $g$ function is continuous, than the function $f$ is continuous.
Prove or disprove the following statement. If $g:(0,1) \to (0,1)$ is an increasing function and satisfies
$g(x) > x$ for all $x \in (0,1)$, then there exists a continuous function $f:(0,1) \to \mathbb{R}$ satisfying $f(x) < f(g(x)) $ for all $x \in (0,1)$, but $f$ is not an increasing function.
Let $\alpha>1$ and $f:\left[\frac{1}{\alpha},\alpha\right]\rightarrow \left[\frac{1}{\alpha},\alpha\right]$, a bijective function. If $f^{-1}(x)=\frac{1}{f(x)},\ \forall x\in \left[\frac{1}{\alpha},\alpha\right]$, prove that:
a)$f$ has at least one point of discontinuity;
b)if $f$ is continuous in $1$, then $f$ has an infinity points of discontinuity;
c)there is a function $f$ which satisfies the conditions from the hypothesis and has a finite number of points of dicontinuity.
[i]Radu Mortici [/i]
Let $I \subset \mathbb{R}$ be a nondegenerate interval and $f:I \to \mathbb{R}$ a differentiable function. We denote $J= \left\{ \frac{f(b)-f(a)}{b-a} : a,b \in I, a<b \right\}.$ Prove that:
$a)$ $J$ is an interval;
$b)$ $J \subset f'(I),$ and the set $f'(I) \setminus J$ contains at most two elements;
$c)$ Using parts $a)$ and $b),$ deduce that $f'$ has the intermediate value property.
Let $a_1,a_2,\dots$ be real numbers. Suppose there is a constant $A$ such that for all $n,$
\[\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.\]
Prove there is a constant $B>0$ such that for all $n,$
\[\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.\]
$n$ is a positive integer, $K$ the set of polynoms of real variables $x_1,x_2,...,x_{n+1}$ and $y_1,y_2,...,y_{n+1}$, function $f:K\rightarrow K$ satisfies
\[f(p+q)=f(p)+f(q),\quad f(pq)=f(p)q+pf(q),\quad (\forall)p,q\in K.\]
If $f(x_i)=(n-1)x_i+y_i,\quad f(y_i)=2ny_i$ for all $i=1,2,...,n+1$ and
\[\prod_{i=1}^{n+1}(tx_i+y_i)=\sum_{i=0}^{n+1}p_it^{n+1-i}\]
for any real $t$, prove, that for all $k=1,...,n+1$
\[f(p_{k-1})=kp_k+(n+1)(n+k-2)p_{k-1}\]