Consider a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ Show that: [b]a)[/b] $ f $ is nondecreasing, if $ f+g $ is nondecreasing for any increasing function $ g:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]b)[/b] $ f $ is nondecreasing, if $ f\cdot g $ is nondecreasing for any increasing function $ g:\mathbb{R}\longrightarrow\mathbb{R} . $ [i]Cristian Mangra[/i]

Let $f(x)$ and $g(x)$ be mutually inverse decreasing functions on the interval $(0,\infty)$. Can it hold that $f(x)>g(x)$ for all $x\in(0,\infty)$?

Let $n>1 \in \mathbb{N}$ and $a_1, a_2, ..., a_n$ be a sequence of $n$ natural integers. Let: $$b_1 = \left[\frac{a_2 + \cdots + a_n}{n-1}\right], b_i = \left[\frac{a_1 + \cdots + a_{i-1} + a_{i+1} + \cdots + a_n}{n-1}\right], b_n = \left[\frac{a_1 + \cdots + a_{n-1}}{n-1}\right]$$ Define a mapping $f$ by $f(a_1,a_2, \cdots a_n) = (b_1,b_2,\cdots,b_n)$. a) Let $g: \mathbb{N} \to \mathbb{N}$ be a function such that $g(1)$ is the number of different elements in $f(a_1,a_2, \cdots a_n)$ and $g(m)$ is the number od different elements in $f^m(a_1,a_2, \cdots a_n) = f(f^{m-1}(a_1,a_2, \cdots a_n)); m>1$. Prove that $\exists k_0 \in \mathbb{N}$ s.t. for $m \ge k_0$ the function $g(m)$ is periodic. b) Prove that $\sum_{m=1}^k \frac{g(m)}{m(m+1)} < C$ for all $k \in \mathbb{N}$, where $C$ is a function that doesn't depend on $k$.

We have a machine that has an input and an output. The input is a letter from the finite set $I$ and the output is a lamp that at each moment has one of the colors of the set $C=\{c_1,\dots,c_p\}$. At each moment the machine has an inner state that is one of the $n$ members of finite set $S$. The function $o: S \rightarrow C$ is a surjective function defining that at each state, what color must the lamp be, and the function $t:S \times I \rightarrow S$ is a function defining how does giving each input at each state changes the state. We only shall see the lamp and we have no direct information from the state of the car at current moment. In other words a machine is $M=(S,I,C,o,t)$ such that $S,I,C$ are finite, $t:S \times I \rightarrow S$ , and $o:S \rightarrow C$ is surjective. It is guaranteed that for each two different inner states, there's a sequence of inputs such that the color of the lamp after giving the sequence to the machine at the first state is different from the color of the lamp after giving the sequence to the machine at the second state. (a) The machine $M$ has $n$ different inner states. Prove that for each two different inner states, there's a sequence of inputs of length no more than $n-p$ such that the color of the lamp after giving the sequence to the machine at the first state is different from the color of the lamp after giving the sequence to the machine at the second state. (b) Prove that for a machine $M$ with $n$ different inner states, there exists an algorithm with no more than $n^2$ inputs that starting at any unknown inner state, at the end of the algorithm the state of the machine at that moment is known. Can you prove the above claim for $\frac{n^2}{2}$?

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

Let $f,g:[a,b]\to [0,\infty)$ be two continuous and non-decreasing functions such that each $x\in [a,b]$ we have \[ \int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt. \] Prove that \[ \int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt. \]

Consider the sequence $ a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1$, where $ z\in \mathbb{C}^*$ is given. i) Prove that if $ a_1>2$, then: \[ a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^*\] ii) Prove that if there is a $ k\in \mathbb{N}^*$ such that $ a_k\le 2$, then $ a_1\le 2$.

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

Let $g(x) = ax^2 + bx + c$ a quadratic function with real coefficients such that the equation $g(g(x)) = x$ has four distinct real roots. Prove that there isn't a function $f$: $R--R$ such that $f(f(x)) = g(x)$ for all $x$ real

Let $f : N \to N$ be a function such that $f(f(1995)) = 95, f(xy) = f(x)f(y)$ and $f(x) \le x$ for all $x,y$. Find all possible values of $f(1995)$.

Let $ n$ be a non-negative integer, which ends written in decimal notation on exactly $ k$ zeros, but which is bigger than $ 10^k$. For a $ n$ is only $ k\equal{}k(n)\geq2$ known. In how many different ways (as a function of $ k\equal{}k(n)\geq2$) can $ n$ be written as difference of two squares of non-negative integers at least?

The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$. There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$. Find $n-10m$.

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.

Given function $f(x) = \sin x + \sin \pi x$ and positive number $d$. Prove that there exists real number $p$ such that $|f(x + p) - f(x)| < d$ holds for all real numbers $x$, and the value of $p$ can be arbitrarily large.

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.

Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.

For a given integer $n \ge 2$, find the maximum possible value of $V_n = \sin x_1 \cos x_2 +\sin x_2 \cos x_3 +...+\sin x_n \cos x_1$, where $x_1,x_2,...,x_n$ are real numbers.

If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$? $\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$

Let $a, b, c$ be positive real numbers and let $[x]$ denote the greatest integer that does not exceed the real number $x$. Suppose that $f$ is a function defined on the set of non-negative integers $n$ and taking real values such that $f(0) = 0$ and \[f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1.\] Prove that if $b + c < 1$, there is a real number $k$ such that \[f(n) \leq kn \qquad \text{ for all } n \qquad (1)\] while if $b + c = 1$, there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$. Show that if $b + c = 1$, there may not be a real number $k$ that satisfies $(1).$

The sequence of digits \[123456789101112131415161718192021\ldots\] is obtained by writing the positive integers in order. If the $10^n$th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2) = 2$ because the $100^{\text{th}}$ digit enters the sequence in the placement of the two-digit integer $55$. Find the value of $f(2007)$.

Let $0<a<1$. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous at $x = 0$ such that $f(x) + f(ax) = x,\, \forall x \in \mathbb{R}$

By number-theoretical functions, we will understand integer-valued functions defined on the set of all integers. Are there number-theoretical functions $ f_0(x),f_1(x),f_2(x),\dots$ such that every number theoretical function $ F(x)$ can be uniquely represented in the form $ F(x)\equal{}\sum_{k\equal{}0}^{\infty}a_kf_k(x)$, $ a_0,a_1,a_2,\dots$ being integers?

For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

The function $f: \mathbb{N}\to\mathbb{N}_{0}$ satisfies for all $m,n\in\mathbb{N}$: \[f(m+n)-f(m)-f(n)=0\text{ or }1, \; f(2)=0, \; f(3)>0, \; \text{ and }f(9999)=3333.\] Determine $f(1982)$.

Let $ S$ be a sequence that: \[ \left\{ \begin{array}{cc} S_0\equal{}0\hfill\\ S_1\equal{}1\hfill\\ S_n\equal{}S_{n\minus{}1}\plus{}S_{n\minus{}2}\plus{}F_n& (n>1) \end{array} \right.\] such that $ F_n$ is Fibonacci sequence such that $ F_1\equal{}F_2\equal{}1$. Find $ S_n$ in terms of Fibonacci numbers.