Prove that from any set of one hundred different whole numbers one can choose either one number which is divisible by $100$, or several numbers whose sum is divisible by $100$.

Points on a spherical surface with radius $4$ are colored in $4$ different colors. Prove that there exist two points with the same color such that the distance between them is either $4\sqrt{3}$ or $2\sqrt{6}$. (Distance is Euclidean, that is, the length of the straight segment between the points)

Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.

Show that there is a unique positive integer which consists of the digits $2$ and $5$, having $2005$ digits and divisible by $2^{2005}$.

$(SWE 1)$ Six points $P_1, . . . , P_6$ are given in $3-$dimensional space such that no four of them lie in the same plane. Each of the line segments $P_jP_k$ is colored black or white. Prove that there exists one triangle $P_jP_kP_l$ whose edges are of the same color.

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

Consider an $N\times N$ matrix, where $N$ is an odd positive integer, such that all its entries are $-1,0$ or $1$. Consider the sum of the numbers in every line and every column. Prove that at least two of the $2N$ sums are equal.

In a school there are $ 10$ rooms. Each student from a room knows exactly one student from each one of the other $ 9$ rooms. Prove that the rooms have the same number of students (we suppose that if $ A$ knows $ B$ then $ B$ knows $ A$).

Let $a,b$ be positive real numbers. For any positive integer $n$, denote by $x_n$ the sum of digits of the number $[an+b]$ in it's decimal representation. Show that the sequence $(x_n)_{n\ge 1}$ contains a constant subsequence. [i]Laurentiu Panaitopol[/i]

Let $ n\ge12$ be an integer, and let $ P_1,P_2,...P_n, Q$ be distinct points in a plane. Prove that for some $ i$, at least $ \frac{n}{6}\minus{}1$ of the distances $ P_1P_i,P_2P_i,...P_{i\minus{}1}P_i,P_{i\plus{}1}P_i,...P_nP_i$ are less than $ P_iQ$.

Let $a, b, c, d$ be integers. Show that the product \[(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\] is divisible by $12$.

Find the largest integer $n$ for which there exist $n$ different integers such that none of them are divisible by either of $7,11$ or $13$, but the sum of any two of them is divisible by at least one of $7,11$ and $13$.

Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?

Let $ A$ be a subset of $ \{ 0,1,2,...,1997 \}$ containing more than $ 1000$ elements. Prove that either $ A$ contains a power of $ 2$ (that is, a number of the form $ 2^k$ with $ k\equal{}0,1,2,...)$ or there exist two distinct elements $ a,b \in A$ such that $ a\plus{}b$ is a power of $ 2$.

Let $n$ to be a positive integer. A family $\wp$ of intervals $[i, j]$ with $0 \le i < j \le n$ and $i$, $j$ integers is considered [i]happy[/i] if, for any $I_1 = [i_1, j_1] \in \wp$ and $I_2 = [i_2, j_2] \in \wp$ such that $I_1 \subset I_2$, we have $i_1 = i_2$ [b]or[/b] $j_1 = j_2$. Determine the maximum number of elements of a [i]happy[/i] family.

Six distinct positive integers are randomly chosen between $1$ and $2011;$ inclusive. The probability that some pair of the six chosen integers has a difference that is a multiple of $5 $ is $n$ percent. Find $n.$

Find all positive integers $n$ with the property that the set \[\{n,n+1,n+2,n+3,n+4,n+5\}\] can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

Prove that among $81$ natural numbers whose prime divisors are in the set $\{2, 3, 5\}$ there exist four numbers whose product is the fourth power of an integer.

Let $n\ge 1$ be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of $n$ spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either • places a stone in an empty space, or • removes a stone from a nonempty space $s,$ places a stone in the nearest empty space to the left of $s$ (if such a space exists), and places a stone in the nearest empty space to the right of $s$ (if such a space exists). Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?

The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$

There are $25$ points on the plane, and among any three of them there are two at a distance less than $1$. Prove that there is a circle of radius $1$ containing at least $13$ of these points.

Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).

Let $S$ be a set of 6 integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7$

Let $ n\geq 3 $ be an integer and let $ \mathcal{S} \subset \{1,2,\ldots, n^3\} $ be a set with $ 3n^2 $ elements. Prove that there exist nine distinct numbers $ a_1,a_2,\ldots,a_9 \in \mathcal{S} $ such that the following system has a solution in nonzero integers: \begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 \\ a_4x + a_5 y + a_6 z &=& 0 \\ a_7x + a_8y + a_9z &=& 0. \end{eqnarray*} [i]Marius Cavachi[/i]