Found problems: 4776
A function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has property $ \mathcal{F} , $ if for any real number $ a, $ there exists a $ b<a $ such that $ f(x)\le f(a), $ for all $ x\in (b,a) . $
[b]a)[/b] Give an example of a function with property $ \mathcal{F} $ that is not monotone on $ \mathbb{R} . $
[b]b)[/b] Prove that a continuous function that has property $ \mathcal{F} $ is nondecreasing.
A function $f(n)$ is defined on the set of positive integers is said to be multiplicative if $f(mn)=f(m)f(n)$ whenever $m$ and $n$ have no common factors greater than $1$. Are the following functions multiplicative? Justify your answer.
(a) $g(n)=5^k$ where $k$ is the number of distinct primes which divide $n$.
(b) $h(n)=\begin{cases} 0 & \text{if} \ n \ \text{is divisible by} \ k^2 \ \text{for some integer} \ k>1 \\ 1 & \text{otherwise} \end{cases}$
The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$?
$\textbf{(A) }874\qquad
\textbf{(B) }883\qquad
\textbf{(C) }887\qquad
\textbf{(D) }891\qquad
\textbf{(E) }892\qquad$
Positive real numbers $a$ and $b$ satisfy the following conditions: the function $f(x)=x^3+ax^2+2bx-1$ has three different real roots, while the function $g(x)=2x^2+2bx+a$ doesn't have real roots.
Prove that $a-b>1$.
[i](V. Karamzin)[/i]
Let $ \triangle ABC$ be a right-angled triangle with $ \angle C\equal{}90^\circ$. $ CD$ is the altitude from $ C$ to $ AB$, with $ D$ on $ AB$. $ \omega$ is the circumcircle of $ \triangle BCD$. $ \omega_1$ is a circle situated in $ \triangle ACD$, it is tangent to the segments $ AD$ and $ AC$ at $ M$ and $ N$ respectively, and is also tangent to circle $ \omega$.
(i) Show that $ BD\cdot CN\plus{}BC\cdot DM\equal{}CD\cdot BM$.
(ii) Show that $ BM\equal{}BC$.
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0, a_1, \cdots $ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n \ge 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
Given a sequence {$a_n$} such that $a_1=2$ and for all positive integer $n\geq 2$ $a_{n+1}=\frac{a_n^4+9}{16a_n}$. Prove that $\frac {4}{5}<a_n<\frac {5}{4}$
Let $[z]$ denote the greatest integer not exceeding $z$. Let $x$ and $y$ satisfy the simultaneous equations
\[ \begin{array}{c} y=2[x]+3, \\ y=3[x-2]+5. \end{array} \]If $x$ is not an integer, then $x+y$ is
$\textbf {(A) } \text{an integer} \qquad \textbf {(B) } \text{between 4 and 5} \qquad \textbf {(C) } \text{between -4 and 4} \qquad \textbf {(D) } \text{between 15 and 16} \qquad \textbf {(E) } 16.5$
Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$
Prove that either $f$ is the identity function or $g$ is periodic.
[i]Proposed by Pranjal Srivastava[/i]
Let $c_1,c_2,\ldots,c_{6030}$ be 6030 real numbers. Suppose that for any 6030 real numbers $a_1,a_2,\ldots,a_{6030}$, there exist 6030 real numbers $\{b_1,b_2,\ldots,b_{6030}\}$ such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for $n=1,2,\ldots,6030$. Find $c_{6030}$.
[i]Victor Wang.[/i]
Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$. Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$.
[i]Proposed by Iurie Boreico[/i]
Consider a latticelike packing of translates of a convex region $K$. Let $t$ be the area of the fundamental parallelogram of the lattice defining the packing, and let $t_{\min} (K)$ denote the minimal value of $t$ taken for all latticelike packings. Is there a natural number $N$ such that for any $n>N$ and for any $K$ different from a parallelogram, $nt_{\min} (K)$ is smaller that the area of any convex domain in which $n$ translates to $K$ can be placed without overlapping? (By a [i]latticelike packing[/i] of $K$ we mean a set of nonoverlapping translates of $K$ obtained from $K$ by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]
Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$:
a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$
b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that the following conditions are true for every pair of positive integers $(x, y)$:
$(i)$: $x$ and $f(x)$ have the same number of positive divisors.
$(ii)$: If $x \nmid y$ and $y \nmid x$, then:
$$\gcd(f(x), f(y)) > f(\gcd(x, y))$$
Suppose that
\[f(x)=\{\begin{array}{cc}n,& \qquad n \in \mathbb N , x= \frac 1n\\ \text{} \\x, & \mbox{otherwise}\end{array}\]
[b]i)[/b] In which points, the function has a limit?
[b]ii)[/b] Prove that there does not exist limit of $f$ in the point $x=0.$
Let $g(t)$ and $h(t)$ be real, continuous functions for $t\geq 0.$ Show that any function $v(t)$ satisfying the differential inequality
$$\frac{dv}{dt}+g(t)v \geq h(t),\;\; v(t)=c,$$
satisfies the further inequality $v(t)\geq u(t),$ where
$$\frac{du}{dt}+g(t)u = h(t),\;\; u(t)=c.$$
From this, conclude that for sufficiently small $t>0,$ the solution of
$$\frac{dv}{dt}+g(t)v = v^2 ,\;\; v(t)=c$$
may be written
$$v=\max_{w(t)} \left( c e^{- \int_{0}^{t} |g(s)-2w(s)| \, ds} -\int_{0}^{t} e^{-\int_{0}^{t} |g(s')-2w(s')| \, ds'} w(s)^{2} ds \right),$$
where the maximum is over all continuous functions $w(t)$ defined over some $t$-interval $[0,t_0 ].$
Let $n$ be an integer bigger than $0$. Let $\mathbb{A}= ( a_1,a_2,...,a_n )$ be a set of real numbers. Find the number of functions $f:A \rightarrow A$ such that $f(f(x))-f(f(y)) \ge x-y$ for any $x,y \in \mathbb{A}$, with $x>y$.
Determine all positive integers $n$ for which there exists a polynomial $f(x)$ with real coefficients, with the following properties:
(1) for each integer $k$, the number $f(k)$ is an integer if and only if $k$ is not divisible by $n$;
(2) the degree of $f$ is less than $n$.
[i](Hungary) Géza Kós[/i]
Suppose that every integer has been given one of the colours red, blue, green or yellow. Let $x$ and $y$ be odd integers so that $|x| \neq |y|$. Show that there are two integers of the same colour whose difference has one of the following values: $x,y,x+y$ or $x-y$.
A function $f_n(x)\ (n=0,\ 1,\ 2,\ 3,\ \cdots)$ satisfies the following conditions:
(i) $f_0(x)=e^{2x}+1$.
(ii) $f_n(x)=\int_0^x (n+2t)f_{n-1}(t)dt-\frac{2x^{n+1}}{n+1}\ (n=1,\ 2,\ 3,\ \cdots).$
Find $\sum_{n=1}^{\infty} f_n'\left(\frac 12\right).$
Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$
Show that $m_n \to \frac{1}{2}$, as $n \to \infty.$
If a, b, and c are non-negative real numbers satisfying $a + b + c = 400$, fi nd the maximum possible value of $\sqrt{2a+b}+\sqrt{2b+c}+\sqrt{2c+a}$.
Find all functions $f:\mathbb R \to \mathbb R$ such that for all $x,y\in \mathbb R$
$f(x+f(y))=2x+2f(y+1)$
Let $ F:\mathbb{R}\longrightarrow\mathbb{R} $ be a primitive with $ F(0)=0 $ of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $ f(x)=\sin (x^2) , $ and let be a sequence $ \left( a_n \right)_{n\ge 0} $ with $ a_0\in (0,1) $ and defined as $ a_{n}=a_{n-1}-F\left( a_{n-1} \right) . $
Calculate $ \lim_{n\to\infty } a_n\sqrt{n} . $
[i]Florian Dumitrel[/i]