Found problems: 4776
Show that if $f: [0,1]\rightarrow [0,1]$ is a continous function and it has topological transitivity then periodic points of $f$ are dense in $[0,1]$.
Topological transitivity means there for every open sets $U$ and $V$ there is $n>0$ such that $f^n(U)\cap V\neq \emptyset$.
Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]
Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$
For real numbers $a,b,c,d$ not all equal to $0$ , define a real function $f(x) = a +b\cos{2x} + c\sin{5x} +d \cos{8x}$. Suppose $f(t) = 4a$ for some real $t$. prove that there exist a real number $s$ s.t. $f(s)<0$
Let be areal number $ r, $ a nonconstant and continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ with period $ T $ and $ F $ be its primitive having $ F(0)=0. $ Define the funtion $ g:\mathbb{R}\longrightarrow\mathbb{R} $ as
$$ g(x)=\left\{\begin{matrix} f(1/x), & x\neq 0 \\ r, & x=0 \end{matrix}\right. $$
Prove that:
[b]a)[/b] the image of $ f $ is closed.
[b]b)[/b] $ g $ has the intermediate value property if and only if $ r\in f\left(\mathbb{R}\right) . $
[b]c)[/b] $ g $ is primitivable if and only if $ r=\frac{F(T)}{T} . $
A function $f$ satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.
Find all positive integers $m$ and $n$ such that the inequality:
\[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \]
is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.
Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that
$$
\sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0.
$$
Determine the maximum possible value of
$$
\sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i).
$$
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x + f(y)) + f(x - f(y)) = x.$$
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
A geometric sequence $ (a_n)$ has $ a_1\equal{}\sin{x}, a_2\equal{}\cos{x},$ and $ a_3\equal{}\tan{x}$ for some real number $ x$. For what value of $ n$ does $ a_n\equal{}1\plus{}\cos{x}$?
$ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$
For how many positive real numbers $a$ has the equation $a^2x^2 + ax+1-7a^2 = 0$ two distinct integer roots?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of above}
$
The function $f$ is defined by $$f(x) = \Big\lfloor \lvert x \rvert \Big\rfloor - \Big\lvert \lfloor x \rfloor \Big\rvert$$for all real numbers $x$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$. What is the range of $f$?
$\textbf{(A) } \{-1, 0\} \qquad\textbf{(B) } \text{The set of nonpositive integers} \qquad\textbf{(C) } \{-1, 0, 1\}$
$\textbf{(D) } \{0\} \qquad\textbf{(E) } \text{The set of nonnegative integers} $
Let $ \{a_k\}$ be a sequence of integers such that $ a_1 \equal{} 1$ and $ a_{m \plus{} n} \equal{} a_m \plus{} a_n \plus{} mn$, for all positive integers $ m$ and $ n$. Then $ a_{12}$ is
$ \textbf{(A)}\ 45 \qquad \textbf{(B)}\ 56 \qquad \textbf{(C)}\ 67 \qquad \textbf{(D)}\ 78 \qquad \textbf{(E)}\ 89$
Find all functions $ f:\mathbb{R}\to\mathbb{R} $ such that $ f(f(x)^2+f(y)) = xf(x)+y $,$ \forall x,y\in R $.
There exist $4$ positive integers $a,b,c,d$ such that $abcd \neq 1$ and each pair of them have a GCD of $1$. Two functions $f,g : \mathbb{N} \rightarrow \{0,1\}$ are multiplicative functions such that for each positive integer $n$ we have :
$$f(an+b)=g(cn+d)$$
Prove that at least one of the followings hold.
$i)$ for each positive integer $n$ we have $f(an+b)=g(cn+d)=0$
$ii)$ There exists a positive integer $k$ such that for all $n$ where $(n,k)=1$ we have $g(n)=f(n)=1$
(Function $f$ is multiplicative if for any natural numbers $a,b$ we have $f(ab)=f(a)f(b)$)
Proposed by [i]Navid Safaii[/i]
The function $f$ is defined by $f\left(x\right)=x^2+3x$. Find the product of all solutions of the equation $f\left(2x-1\right)=6$.
Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the
$$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$
inequality for all $x,y\in I$ and $t\in [0,1]$. Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$-Lipschitz property, that is
$$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$
Two particles move along the edges of equilateral triangle $ \triangle ABC$ in the direction
\[ A\rightarrow B\rightarrow C\rightarrow A
\]starting simultaneously and moving at the same speed. One starts at $ A$, and the other starts at the midpoint of $ \overline{BC}$. The midpoint of the line segment joining the two particles traces out a path that encloses a region $ R$. What is the ratio of the area of $ R$ to the area of $ \triangle ABC$?
$ \textbf{(A)}\ \frac {1}{16}\qquad \textbf{(B)}\ \frac {1}{12}\qquad \textbf{(C)}\ \frac {1}{9}\qquad \textbf{(D)}\ \frac {1}{6}\qquad \textbf{(E)}\ \frac {1}{4}$
Let $ABC$ a triangle be given with $BC = a$, $CA = b$, $AB = c$ ($a \neq b \neq c \neq a$). In plane ($ABC$) take the points $A'$, $B'$, $C'$ such that:
[b]I.[/b] The pairs of points $A$ and $A'$, $B$ and $B'$, $C$ and $C'$ either all lie in one side either all lie in different sides under the lines $BC$, $CA$, $AB$ respectively;
[b]II.[/b] Triangles $A'BC$, $B'CA$, $C'AB$ are similar isosceles triangles.
Find the value of angle $A'BC$ as function of $a, b, c$ such that lengths $AA', BB', CC'$ are not sides of an triangle. (The word "triangle" must be understood in its ordinary meaning: its vertices are not collinear.)
We say that a natural number $ n\ge 4 $ is [i]unusual[/i] if, for any $ n\times n $ array of real numbers, the sum of the numbers from any $ 3\times 3 $ compact subarray is negative, and the sum of the numbers from any $ 4\times 4 $ compact subarray is positive.
Find all unusual numbers.
If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, and for all $a$ and $b$ \[f(a+b)+f(a-b)=2f(a)+2f(b),\] then for all $x$ and $y$
$\textbf{(A) }f(0)=1\qquad\textbf{(B) }f(-x)=-f(x)\qquad$
$\textbf{(C) }f(-x)=f(x)\qquad\textbf{(D) }f(x+y)=f(x)+f(y)\qquad$
$\textbf{(E) }\text{there is a positive real number }T\text{ such that }f(x+T)=f(x)$
Let $f(x)=ax^2+bx+c$ be a quadratic function, the graph of which doesn't intersect the x-axis. Prove that
$$a(2a+3b+6c)>0.$$
Each vertex of a regular $11$-gon is colored black or gold. All possible triangles are formed using these vertices. Prove that there are either two congruent triangles with three black vertices or two congruent triangles with three gold vertices.
a) Find all multiplicative functions $f: \mathbb Z_{p}^{*}\longrightarrow\mathbb Z_{p}^{*}$ (i.e. that $\forall x,y\in\mathbb Z_{p}^{*}$, $f(xy)=f(x)f(y)$.)
b) How many bijective multiplicative does exist on $\mathbb Z_{p}^{*}$
c) Let $A$ be set of all multiplicative functions on $\mathbb Z_{p}^{*}$, and $VB$ be set of all bijective multiplicative functions on $\mathbb Z_{p}^{*}$. For each $x\in \mathbb Z_{p}^{*}$, calculate the following sums :\[\sum_{f\in A}f(x),\ \ \sum_{f\in B}f(x)\]