The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? ${ \textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}}\ 991\qquad\textbf{(E)}\ 999 $

Prove that for any positive integer $n\geq 2$ we have that \[\sum_{k=2}^n \lfloor \sqrt[k]{n}\rfloor=\sum_{k=2}^n\lfloor\log_{k}n\rfloor.\]

Prove that in a plane, arbitrary $ n$ points can be overlapped by discs that the sum of all the diameters is less than $ n$, and the distances between arbitrary two are greater than $ 1$. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have common points are zero)

The game of circulate is played with a deck of $kn$ cards each with a number in $1,2,\ldots,n$ such that there are $k$ cards with each number. First, $n$ piles numbered $1,2,\ldots,n$ of $k$ cards each are dealt out face down. The player then flips over a card from pile $1$, places that card face up at the bottom of the pile, then next flips over a card from the pile whose number matches the number on the card just flipped. The player repeats this until he reaches a pile in which every card has already been flipped and wins if at that point every card has been flipped. Hamster has grown tired of losing every time, so he decides to cheat. He looks at the piles beforehand and rearranges the $k$ cards in each pile as he pleases. When can Hamster perform this procedure such that he will win the game? [i]Brian Hamrick.[/i]

$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions: [b]I.[/b] $f(1) = 2$ [b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$ Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.

The sequence of polynomials $(a_n)$ is defined by $a_0=0$, $ a_1=x+2$ and $a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2}$ for $n>1$. (a) Show for all positive integers $k,m$: if $k$ divides $m$ then $a_k$ divides $a_m$. (b) Find all positive integers $n$ such that the sum of the roots of polynomial $a_n$ is an integer.

Let $ n \equal{} 2k \minus{} 1$ where $ k \geq 6$ is an integer. Let $ T$ be the set of all $ n\minus{}$tuples $ (x_1, x_2, \ldots, x_n)$ where $ x_i \in \{0,1\}$ $ \forall i \equal{} \{1,2, \ldots, n\}$ For $ x \equal{} (x_1, x_2, \ldots, x_n) \in T$ and $ y \equal{} (y_1, y_2, \ldots, y_n) \in T$ let $ d(x,y)$ denote the number of integers $ j$ with $ 1 \leq j \leq n$ such that $ x_i \neq x_j$, in particular $ d(x,x) \equal{} 0.$ Suppose that there exists a subset $ S$ of $ T$ with $ 2^k$ elements that has the following property: Given any element $ x \in T,$ there is a unique element $ y \in S$ with $ d(x, y) \leq 3.$ Prove that $ n \equal{} 23.$

Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.

Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.

Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.

Three coins lie on integer points on the number line. A move consists of choosing and moving two coins, the first one $ 1$ unit to the right and the second one $ 1$ unit to the left. Under which initial conditions is it possible to move all coins to one single point?

Let $P(x)$ be a nonzero polynomial with integer coefficients. Let $a_{0}=0$ and for $i \ge 0$ define $a_{i+1}=P(a_{i})$. Show that $\gcd ( a_{m}, a_{n})=a_{ \gcd (m, n)}$ for all $m, n \in \mathbb{N}$.

For a positive integer $ n,$ $ a_{n}\equal{}n\sqrt{5}\minus{} \lfloor n\sqrt{5}\rfloor$. Compute the maximum value and the minimum value of $ a_{1},a_{2},\ldots ,a_{2009}.$

Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties: 1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$; 2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$; Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.

Let $ \{x_n\}_{n\geq 1}$ be a sequences, given by $ x_1 \equal{} 1$, $ x_2 \equal{} 2$ and \[ x_{n \plus{} 2} \equal{} \frac { x_{n \plus{} 1}^2 \plus{} 3 }{x_n} . \] Prove that $ x_{2008}$ is the sum of two perfect squares.

Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties: 1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$; 2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$; Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.

Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that \[a_{m^2+n^2}=a_m^2+a_n^2 \] for all integers $m,n\ge 0$.

The sequence $ \{a_n\}$ of integers is defined by \[ a_1 \equal{} 2, a_2 \equal{} 7 \] and \[ \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2. \] Prove that $ a_n$ is odd for all $ n > 1.$

Let $T$ a set with 2007 points on the plane, without any 3 collinear points. Let $P$ any point which belongs to $T$. Prove that the number of triangles that contains the point $P$ inside and its vertices are from $T$, is even.

Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.

We have $n$ countries. Each country have $m$ persons who live in that country ($n>m>1$). We divide $m \cdot n$ persons into $n$ groups each with $m$ members such that there don't exist two persons in any groups who come from one country. Prove that one can choose $n$ people into one class such that they come from different groups and different countries.

Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows: (1). $ \dfrac{1}{2} \in \mathbb{H}$, (2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1\plus{}x} \in \mathbb{H}$ and also $ \dfrac{x}{1\plus{}x} \in \mathbb{H}$. Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.

In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

Let $n$ be a fixed positive odd integer. Take $m+2$ [b]distinct[/b] points $P_0,P_1,\ldots ,P_{m+1}$ (where $m$ is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied: 1) $P_0=(0,1),P_{m+1}=(n+1,n)$, and for each integer $i,1\le i\le m$, both $x$- and $y$- coordinates of $P_i$ are integers lying in between $1$ and $n$ ($1$ and $n$ inclusive). 2) For each integer $i,0\le i\le m$, $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and is parallel to the $y$-axis if $i$ is odd. 3) For each pair $i,j$ with $0\le i<j\le m$, line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most $1$ point. Determine the maximum possible value that $m$ can take.

For any natural number $n > 1$ write the finite decimal expansion of $\frac{1}{n}$ (for example we write $\frac{1}{2}=0.4\overline{9}$ as its infinite decimal expansion not $0.5)$. Determine the length of non-periodic part of the (infinite) decimal expansion of $\frac{1}{n}$.