Let $ M$ be the set of those positive integers which are not divisible by $ 3$. The sum of $ 2n$ consecutive elements of $ M$ is $ 300$. Determine $ n$.

We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?

Let $A_1,A_2,\ldots,A_n$ be subsets of a finite set $S$ such that $|A_j|=8$ for each $j$. For a subset $B$ of $S$ let $F(B)=\{j \mid 1\le j\le n \ \ \text{and} \ A_j \subset B\}$. Suppose for each subset $B$ of $S$ at least one of the following conditions holds [list][b](a)[/b] $|B| > 25$, [b](b)[/b] $F(B)={\O}$, [b](c)[/b] $\bigcap_{j\in F(B)} A_j \neq {\O}$.[/list] Prove that $A_1\cap A_2 \cap \cdots \cap A_n \neq {\O}$.

Prove that every positive integer $n$, except a finite number of them, can be represented as a sum of $2004$ positive integers: $n=a_1+a_2+\cdots +a_{2004}$, where $1\le a_1<a_2<\cdots <a_{2004}$, and $a_i \mid a_{i+1}$ for all $1\le i\le 2003$. [i]Chen Yonggao[/i]

For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \in \mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \in \mathbb{N}$, such that $k \mid m^{h} - 1, m \mid n^{\frac{m^{h}-1}{k}} + 1$.

A positive integer is written on a blackboard. Players $A$ and $B$ play the following game: in each move one has to choose a proper divisor $m$ of the number $n$ written on the blackboard ($1<m<n$) and replaces $n$ with $n-m$. Player $A$ makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player $B$?

Let $n$ be positive integer. Define a sequence $\{a_k\}$ by \[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\] (1) Find $a_2$ and $a_3$. (2) Find the general term $a_k$. (3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$. 50 points

On a planet there are $3\times2005!$ aliens and $2005$ languages. Each pair of aliens communicates with each other in exactly one language. Show that there are $3$ aliens who communicate with each other in one common language.

Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties: (1) $f(0)=0$ and $f(1)=1$; and. (2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

Prove that if $P(x) = (x-a)^kQ(x)$, where $k$ is a positive integer, $a$ is a nonzero real number, $Q(x)$ is a nonzero polynomial, then $P(x)$ has at least $k + 1$ nonzero coefficients.

In a party among any four persons there are three people who are mutual acquaintances or mutual strangers. Prove that all the people can be separated into two groups $A$ and $B$ such that in $A$ everybody knows everybody else and in $B$ nobody knows anybody else.

Fix an integer $k>2$. Two players, called Ana and Banana, play the following game of numbers. Initially, some integer $n \ge k$ gets written on the blackboard. Then they take moves in turn, with Ana beginning. A player making a move erases the number $m$ just written on the blackboard and replaces it by some number $m'$ with $k \le m' < m$ that is coprime to $m$. The first player who cannot move anymore loses. An integer $n \ge k $ is called good if Banana has a winning strategy when the initial number is $n$, and bad otherwise. Consider two integers $n,n' \ge k$ with the property that each prime number $p \le k$ divides $n$ if and only if it divides $n'$. Prove that either both $n$ and $n'$ are good or both are bad.

We define a sequence $a_n$ so that $a_0=1$ and \[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + d & \textrm{ otherwise. } \end{cases} \] for all postive integers $n$. Find all positive integers $d$ such that there is some positive integer $i$ for which $a_i=1$.

Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$ Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$ [i]Edited by orl.[/i]

Let $n$ be a positive odd integer and let $\theta$ be a real number such that $\theta/\pi$ is irrational. Set $a_{k}=\tan(\theta+k\pi/n),\ k=1,2\dots,n.$ Prove that \[\frac{a_{1}+a_{2}+\cdots+a_{n}}{a_{1}a_{2}\cdots a_{n}}\] is an integer, and determine its value.

We put pebbles on some unit squares of a $2013 \times 2013$ chessboard such that every unit square contains at most one pebble. Determine the minimum number of pebbles on the chessboard, if each $19\times 19$ square formed by unit squares contains at least $21$ pebbles.

Show that every natural number $n\leq2^{1\;000\;000}$ can be obtained first with 1 doing less than $1\;100\;000$ sums; more precisely, there is a finite sequence of natural numbers $x_0,\ x_1,\dots,\ x_k\mbox{ with }k\leq1\;100\;000,\ x_0=1,\ x_k=n$ such that for all $i=1,\ 2,\dots,\ k$ there exist $r,\ s$ with $0\leq{r}\leq{s}<i$ such that $x_i=x_r+x_s$.

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic. [i]L. Lovasz, J. Pelikan[/i]

Let $A$ be an infinite set of positive integers such that every $n \in A$ is the product of at most $1987$ prime numbers. Prove that there is an infinite set $B \subset A$ and a number $p$ such that the greatest common divisor of any two distinct numbers in $B$ is $p.$

Let $ n\geqslant 3$ and $ P$ a convex $ n$-gon. Show that $ P$ can be, by $ n \minus{} 3$ non-intersecting diagonals, partitioned in triangles such that the circumcircle of each triangle contains the whole area of $ P$. Under which conditions is there exactly one such triangulation?

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 $a_2=0$.

Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and \[\gcd(P(0), P(1), P(2), \ldots ) = 1.\] Show there are infinitely many $n$ such that \[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]

Let a function $g:\mathbb{N}_0\to\mathbb{N}_0$ satisfy $g(0)=0$ and $g(n)=n-g(g(n-1))$ for all $n\ge 1$. Prove that: a) $g(k)\ge g(k-1)$ for any positive integer $k$. b) There is no $k$ such that $g(k-1)=g(k)=g(k+1)$.