Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$: \[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]

Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$

For an arbitrary real number $ x$, $ \lfloor x\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \displaystyle m\minus{}\left\lfloor \frac{m}{2005}\right\rfloor\equal{}2005$.

Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P\rfloor$.

Let $r,n$ be positive integers. For a set $A$, let ${A \choose r}$ denote the family of all $r$-element subsets of $A$. Prove that if $A$ is infinite and $f : {A \choose r} \to {1,2,...,n}$ is any function, then there exists an infinite subset $B$ of $A$ such that $f(X) = f(Y)$ for all $X,Y \in {B \choose r}$.

Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies: \[f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}\]

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

Find the number of extrema of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\prod_{j=1}^n (x-j)^j, $$ where $ n $ is a natural number.

Find all positive integers $k$ for which there exists a nonlinear function $f:\mathbb{Z} \rightarrow\mathbb{Z}$ such that the equation $$f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k}$$ holds for any integers $a,b,c$ satisfying $a+b+c=0$ (not necessarily distinct). [i]Evan Chen[/i]

Two functions are defined by equations: $f(a) = a^2 + 3a + 2$ and $g(b, c) = b^2 - b + 3c^2 + 3c$. Prove that for any positive integer $a$ there exist positive integers $b$ and $c$ such that $f(a) = g(b, c)$.

Define $ x \heartsuit y$ to be $ |x\minus{}y|$ for all real numbers $ x$ and $ y$. Which of the following statements is [b]not[/b] true? $\textbf{(A)}\ x \heartsuit y \equal{} y \heartsuit x \text{ for all } x \text{ and } y$ $\textbf{(B)}\ 2(x \heartsuit y) \equal{} (2x) \heartsuit (2y) \text{ for all } x \text{ and } y$ $\textbf{(C)}\ x \heartsuit 0 \equal{} x \text{ for all } x$ $\textbf{(D)}\ x \heartsuit x \equal{} 0 \text{ for all } x$ $\textbf{(E)}\ x \heartsuit y > 0 \text{ if } x \ne y$

Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and denominator have a sum of $1000$.

Find the number of functions defined on positive real numbers such that $ f\left(1\right) \equal{} 1$ and for every $ x,y\in \Re$, $ f\left(x^{2} y^{2} \right) \equal{} f\left(x^{4} \plus{} y^{4} \right)$. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

Determine the number of possible values for the determinant of $A$, given that $A$ is a $n\times n$ matrix with real entries such that $A^3 - A^2 - 3A + 2I = 0$, where $I$ is the identity and $0$ is the all-zero matrix.

Let $S_{n}=\{1,2,...,n\}$. How many functions $f:S_{n} \rightarrow S_{n}$ satisfy $f(k) \leq f(k+1)$ and $f(k)=f(f(k+1))$ for $k <n?$

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)-f(y))+2f(xy)=x^2f(x)+f(y^2)$$ for all real numbers $x,y$.

What is the value of the sum \[ \sum_z \frac{1}{{\left|1 - z\right|}^2} \, , \] where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?

There is a connected network with $ 2008$ computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the $ 1st$ move hacker selects one computer and hacks it, on the $ 2nd$ move administrator selects another computer and protects it. Then on every $ 2k\plus{}1th$ move hacker hacks one more computer(if he can) which wasn't protected by the administrator and is directly connected (with an edge) to a computer which was hacked by the hacker before and on every $ 2k\plus{}2th$ move administrator protects one more computer(if he can) which wasn't hacked by the hacker and is directly connected (with an edge) to a computer which was protected by the administrator before for every $ k>0$. If both of them can't make move, the game ends. Determine the maximum number of computers which the hacker can guarantee to hack at the end of the game.

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.$

Does there exist any function $f: \mathbb{R}^+ \to \mathbb{R}$ such that for every positive real number $x,y$ the following is true : $$f(xf(x)+yf(y)) = xy$$

Let $S$ denote the set of words $W = w_1w_2\ldots w_n$ of any length $n\ge0$ (including the empty string $\lambda$), with each letter $w_i$ from the set $\{x,y,z\}$. Call two words $U,V$ [i]similar[/i] if we can insert a string $s\in\{xyz,yzx,zxy\}$ of three consecutive letters somewhere in $U$ (possibly at one of the ends) to obtain $V$ or somewhere in $V$ (again, possibly at one of the ends) to obtain $U$, and say a word $W$ is [i]trivial[/i] if for some nonnegative integer $m$, there exists a sequence $W_0,W_1,\ldots,W_m$ such that $W_0=\lambda$ is the empty string, $W_m=W$, and $W_i,W_{i+1}$ are similar for $i=0,1,\ldots,m-1$. Given that for two relatively prime positive integers $p,q$ we have \[\frac{p}{q} = \sum_{n\ge0} f(n)\left(\frac{225}{8192}\right)^n,\]where $f(n)$ denotes the number of trivial words in $S$ of length $3n$ (in particular, $f(0)=1$), find $p+q$. [i]Victor Wang[/i]

The maximum value of function $f(x)=\sqrt{x^4-3x^2-6x+13}-\sqrt{x^4-x^2+1}$ is________.

2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.

Let $n$ be a positive integer. (1) For a positive integer $k$ such that $1\leq k\leq n$, Show that : \[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\] (2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$ If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$ (3) Find $\lim_{n\to\infty} S_n.$

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.