Found problems: 4776
Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have
\[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]
Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions:
(i) $a,b\in\{1,2,\ldots,2013\}$;
(ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.
$ABCD$ is a quadrilateral with $AD=BC$. If $\angle ADC$ is greater than $\angle BCD$, prove that $AC>BD$.
Let $ X $ be the power set of set of $ \{ 0\}\cup\mathbb{N} , $ and let be a function $ d:X^2\longrightarrow\mathbb{R} $ defined as
$$ d(U,V)=\sum_{n\in\mathbb{N}}\frac{\chi_U (n) +\chi_V (n) -2\chi_{U\cap V} (n)}{2} , $$
where $ \chi_W (n)=\left\{ \begin{matrix} 1,& n\in W\\ 0,& n\not\in W \end{matrix} \right. ,\quad\forall W\in X,\forall n\in\mathbb{N} . $
[b]a)[/b] Prove that there exists an unique $ V' $ such that $ \lim_{k\to\infty} d\left( \{ k+i|i\in\mathbb{N}\} , V'\right) =0. $
[b]b)[/b] Demonstrate that for all $ V\in X $ there exists a $ v\in\mathbb{N} $ with $ d\left( \left\{ \frac{3}{2} -\frac{1}{2}(-1)^{v} \right\} , V \right) >\frac{1}{k} . $
[b]c)[/b] Let $ f: X\longrightarrow X,\quad f(X)=\left\{ 1+x|x\in X\right\} . $ Calculate $ d\left( f(A),f(B) \right) $ in terms of $ d(A,B) $ and prove that $ f $ admits an unique fixed point.
If $ \log(xy^3)\equal{}1$ and $ \log(x^2y)\equal{}1$, what is $ \log(xy)$?
$ \textbf{(A)}\ \minus{}\!\frac{1}{2} \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{3}{5} \qquad
\textbf{(E)}\ 1$
a) There is an infinite sequence of $0,1$, like $\dots,a_{-1},a_{0},a_{1},\dots$ (i.e. an element of $\{0,1\}^{\mathbb Z}$). At each step we make a new sequence. There is a function $f$ such that for each $i$, $\mbox{new }a_{i}=f(a_{i-100},a_{i-99},\dots,a_{i+100})$. This operation is mapping $F: \{0,1\}^{\mathbb Z}\longrightarrow\{0,1\}^{\mathbb Z}$. Prove that if $F$ is 1-1, then it is surjective.
b) Is the statement correct if we have an $f_{i}$ for each $i$?
Let $P(x)$ be a nonzero polynomial of degree $n>1$ with nonnegative coefficients such that function $y=P(x)$ is odd. Is that possible thet for some pairwise distinct points $A_{1}, A_{2}, \dots A_{n}$ on the graph $G: y = P(x)$ the following conditions hold: tangent to $G$ at $A_{1}$ passes through $A_{2}$, tangent to $G$ at $A_{2}$ passes through $A_{3}$, $\dots$, tangent to $G$ at $A_{n}$ passes through $A_{1}$?
Given a circle of radius $r$ and a tangent line $\ell$ to the circle through a given point $P$ on the circle. From a variable point $R$ on the circle, a perpendicular $RQ$ is drawn to $\ell$ with $Q$ on $\ell$. Determine the maximum of the area of triangle $PQR$.
Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.
An $n$-term sequence $(x_1, x_2, \ldots, x_n)$ in which each term is either 0 or 1 is called a [i]binary sequence of length [/i]$n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.
Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.
Let $ I $ be a real interval, and $ f:I\longrightarrow\mathbb{R} $ be a continuous function. Prove that $ f $ is monotone if and only if $ \min(\left( f(a),f(b)\right) \le\frac{1}{b-a}\int_a^b f(x)dx \le\max\left( f(a),f(b) \right) , $ for any distinct $ a,b\in I. $
Let $f:[0,1]\rightarrow [0,1]$ a continuous function in $0$ and in $1$, which has one-side limits in any point and $f(x-0)\le f(x)\le f(x+0),\ (\forall)x\in (0,1)$. Prove that:
a)for the set $A=\{x\in [0,1]\ |\ f(x)\ge x\}$, we have $\sup A\in A$.
b)there is $x_0\in [0,1]$ such that $f(x_0)=x_0$.
[i]Mihai Piticari[/i]
Let $\mathcal{F}$ be the set of the differentiable functions $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(x) \ge f(x+ \sin x)$ for any $x \in \mathbb{R}.$
a) Prove that there exist nonconstant functions in $\mathcal{F}.$
b) Prove that if $f \in \mathcal{F},$ then the set of solutions of the equation $f'(x)=0$ is infinite.
The families of second degree functions $f_m, g_m: R\to R, $ are considered , $f_m (x) = (m^2 + 1) x^2 + 3mx + m^2 - 1$, $g_m (x) = m^2x^2 + mx - 1$, where $m$ is a real nonzero parameter.
Show that, for any function $h$ of the second degree with the property that $g_m (x) \le h (x) \le f_m (x)$ for any real $x$, there exists $\lambda \in [0, 1]$ which verifies the condition $h (x) = \lambda f_m (x) + (1- \lambda) g_m (x)$, whatever real $x$ is.
A calculator is broken so that the only keys that still work are the $ \sin$, $ \cos$, and $ \tan$ buttons, and their inverses (the $ \arcsin$, $ \arccos$, and $ \arctan$ buttons). The display initially shows $ 0$. Given any positive rational number $ q$, show that pressing some finite sequence of buttons will yield the number $ q$ on the display. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.
Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.
Let $\mathbb{N}$ be the set of positive integers. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that for every positive integer $n \in \mathbb{N}$
$$ f(n) -n<2021 \quad \text{and} \quad f^{f(n)}(n) =n$$
Prove that $f(n)=n$ for infinitely many $n \in \mathbb{N}$
Let $ \overline{AB}$ be a diameter of circle $ \omega$. Extend $ \overline{AB}$ through $ A$ to $ C$. Point $ T$ lies on $ \omega$ so that line $ CT$ is tangent to $ \omega$. Point $ P$ is the foot of the perpendicular from $ A$ to line $ CT$. Suppose $ AB \equal{} 18$, and let $ m$ denote the maximum possible length of segment $ BP$. Find $ m^{2}$.
Prove that for all positive real numbers $a,b,c$ the following inequality takes place
\[ \frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} . \]
[i]Laurentiu Panaitopol, Romania[/i]
Let be a differentiable function $ f:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} , $ and a primitive $ F:\mathbb{R}_{> 0}\longrightarrow\mathbb{R}_{> 0} $ of it such that $ F=f+f\cdot f. $ Show that:
[b]a)[/b] $ f $ is nondecreasing.
[b]b)[/b] $ \lim_{x\to\infty } f(x)/x =1/2 $
[i]Vasile Solovăstru[/i]
Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.
Does there exist a function $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+\sin y$, for all reals $x,y$ ?
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
Find all functions $f:\mathbb R\to\mathbb R$ so that the following relation holds for all $x, y\in\mathbb R$.
$$f(f(x)f(y)-1) = xy - 1$$