Given a number $n\in\mathbb{Z}^+$ and let $S$ denotes the set $\{0,1,2,...,2n+1\}$. Consider the function $f:\mathbb{Z}\times S\to [0,1]$ satisfying two following conditions simultaneously: i) $f(x,0)=f(x,2n+1)=0\forall x\in\mathbb{Z}$; ii) $f(x-1,y)+f(x+1,y)+f(x,y-1)+f(x,y+1)=1$ for all $x\in\mathbb{Z}$ and $y\in\{1,2,3,...,2n\}$. Let $F$ be the set of such functions. For each $f\in F$, let $v(f)$ be the set of values of $f$. a) Proof that $|F|=\infty$. b) Proof that for each $f\in F$ then $|v(f)|<\infty$. c) Find the maximum value of $|v(f)|$ for $f\in F$.

Let $X=\left\{1,2,3,4\right\}$. Consider a function $f:X\to X$. Let $f^1=f$ and $f^{k+1}=\left(f\circ f^k\right)$ for $k\geq1$. How many functions $f$ satisfy $f^{2014}\left(x\right)=x$ for all $x$ in $X$? $\text{(A) }9\qquad\text{(B) }10\qquad\text{(C) }12\qquad\text{(D) }15\qquad\text{(E) }18$

Find all continuous functions $f:[0,1]\to [0,1]$ for which there exists a positive integer $n$ such that $f^{n}(x)=x$ for $x \in [0,1]$ where $f^{0} (x)=x$ and $f^{k+1}=f(f^{k}(x))$ for every positive integer $k$.

In the following, let $N_0$ denotes the set of non-negative integers. Find all polynomials $P(x)$ that fulfill the following two properties: (1) All coefficients of $P(x)$ are from $N_0$. (2) Exists a function $f : N_0 \to N_0$ such as $f (f (f (n))) = P (n)$ for all $n \in N_0$.

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]

A real-valued function $ f$ satisfies for all reals $ x$ and $ y$ the equality \[ f (xy) \equal{} f (x)y \plus{} x f (y). \] Prove that this function satisfies for all reals $ x$ and $ y \ne 0$ the equality \[ f\left(\frac{x}{y}\right)\equal{}\frac{f (x)y \minus{} x f (y)}{y^2} \]

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 $ f$ be a function such that for all integers $ x$ and $ y$ applies $ f(x\plus{}y) \equal{} f(x) \plus{} f(y) \plus{} 6xy \plus{} 1$ and $ f(x) \equal{} f(\minus{}x)$. Then $ f(3)$ equals $ \text{(A)}\ 26 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 52 \qquad \text{(D)}\ 53 \qquad \text{(E)}\ 54$

Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for any $x,y\in \mathbb{R}$, there holds \[f(x+f(y))+f(xy)=yf(x)+f(y)+f(f(x)).\]

If $0 \leq a \leq b \leq c \leq d,$ prove that \[a^bb^cc^dd^a \geq b^ac^bd^ca^d.\]

Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$. Show that $|A|\cdot|B|\le|C|^2$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

Let $\mathbb{N} = \{1,2,3, \ldots\}$. Determine if there exists a strictly increasing function $f: \mathbb{N} \mapsto \mathbb{N}$ with the following properties: (i) $f(1) = 2$; (ii) $f(f(n)) = f(n) + n, (n \in \mathbb{N})$.

Let G be an abelian group, $0\leq\varepsilon<1$ and $f : G\to\Bbb R^n$ a function that satisfies the inequality. $$||f(x+y)-f(x)-f(y)|| \leq \varepsilon ||f (y)|| \qquad (x, y)\in G^2$$ Prove that there is an additive function $A : G\to \Bbb R^n$ and a continuous function $\varphi : A (G) \to\Bbb R^n$ such that $f = \varphi\circ A$.

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.

Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively. (a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$ (b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]

A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $ [b]a)[/b] Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $ [b]b)[/b] Prove that if $ f $ is nondecreasing then $ f/\text{id.} $ has a limit at $ \infty , $ and determine it.

Let $ n$ be positive integer, $ A,B\subseteq[0,n]$ are sets of integers satisfying $ \mid A\mid \plus{} \mid B\mid\ge n \plus{} 2.$ Prove that there exist $ a\in A, b\in B$ such that $ a \plus{} b$ is a power of $ 2.$

Find the largest positive value attained by the function \[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \qquad x\ \text{a real number} \] $ \textbf{(A)}\ \sqrt{7}-1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2\sqrt{3} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{55}-\sqrt{5} $

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x, y\in\mathbb{R}$, we have \[f(x+y)+f(x)f(y)=f(xy)+(y+1)f(x)+(x+1)f(y).\]

Find all positive integer solutions of the equation $k!l! = k!+l!+m!.$

The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience. [list][b](a)[/b] Prove that if this is possible for some $n$, then it is also possible for $2n$. [b](b)[/b] Determine all $n$ for which this is possible.[/list]

Find all non-negative real numbers $a$ such that the equation \[ \sqrt[3]{1+x}+\sqrt[3]{1-x}=a \] has at least one real solution $x$ with $0 \leq x \leq 1$. For all such $a$, what is $x$?

Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions: (1) $ f(mn) \equal{} f(m)\plus{}f(n)$, (2) $ f(2008) \equal{} 0$, and (3) $ f(n) \equal{} 0$ for all $ n \equiv 39\pmod {2008}$.

The function $\mu: \mathbb{N}\to \mathbb{C}$ is defined by \[\mu(n) = \sum^{}_{k \in R_{n}}\left( \cos \frac{2k\pi}{n}+i \sin \frac{2k\pi}{n}\right),\] where $R_{n}=\{ k \in \mathbb{N}\vert 1 \le k \le n, \gcd(k, n)=1 \}$. Show that $\mu(n)$ is an integer for all positive integer $n$.

Find the greatest integer less than the number $1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$