A square of side $n$ is formed from $n^2$ unit squares, each colored in red, yellow or green. Find minimal $n$, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or column).

Let $S$ be a set of $ 2 \times 2 $ integer matrices whose entries $a_{ij}(1)$ are all squares of integers and, $(2)$ satisfy $a_{ij} \le 200$. Show that $S$ has more than $ 50387 (=15^4-15^2-15+2) $ elements, then it has two elements that commute.

Let $ \alpha$ be the positive root of the equation $ x^2 \minus{} 1989x \minus{} 1 \equal{} 0.$ Prove that there exist infinitely many natural numbers $ n$ that satisfy the equation: \[ \lfloor \alpha n \plus{} 1989 \alpha \lfloor \alpha n \rfloor \rfloor \equal{} 1989n \plus{} \left( 1989^2 \plus{} 1 \right) \lfloor \alpha n \rfloor.\]

Let $P_1,P_2,\dots,P_n$ be $n$ distinct points over a line in the plane ($n\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\leq{i,j}\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud. For each positive integer $k$, find all the positive integers $n$ such that every possible ($n,k$)-cloud has two mutually exterior tangent circumferences of the same color.

In how many different ways can three knights be placed on a chessboard so that the number of squares attacked would be maximal?

Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the $ 26$ whose product is a square. [hide] I think that the upper limit for such subset is 37.[/hide]

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

There are $ 16$ persons in a company, each of which likes exactly $ 8$ other persons. Show that there exist two persons who like each other.

Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$. Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.

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 $n.$

$101$ distinct numbers are chosen among the integers between $0$ and $1000$. Prove that, among the absolute values ​​of their pairwise differences, there are ten different numbers not exceeding $100$.

Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.

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)

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 $a_i=1$.

There are representants from $n$ different countries sit around a circular table ($n\geq2$), in such way that if two representants are from the same country, then, their neighbors to the right are not from the same country. Find, for every $n$, the maximal number of people that can be sit around the table.

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.

All members of the senate were firstly divided into $S$ senate commissions . According to the rules, no commission has less that $5$ senators and every two commissions have different number of senators. After the first session the commissions were closed and new commissions were opened. Some of the senators now are not a part of any commission. It resulted also that every two senators that were in the same commission in the first session , are not any more in the same commission. [b](a)[/b]Prove that at least $4S+10$ senators were left outside the commissions. [b](b)[/b]Prove that this number is achievable. Albanian National Mathematical Olympiad 2010---12 GRADE Question 5.

Let $n\ge 16$ be an integer, and consider the set of $n^2$ points in the plane: \[ G=\big\{(x,y)\mid x,y\in\{1,2,\ldots,n\}\big\}.\] Let $A$ be a subset of $G$ with at least $4n\sqrt{n}$ elements. Prove that there are at least $n^2$ convex quadrilaterals whose vertices are in $A$ and all of whose diagonals pass through a fixed point.

$(SWE 1)$ 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.

Kati cut two equal regular $ n\minus{}gons$ out of paper. To the vertices of both $ n\minus{}gons$, she wrote the numbers 1 to $ n$ in some order. Then she stabbed a needle through the centres of these $ n\minus{}gons$ so that they could be rotated with respect to each other. Kati noticed that there is a position where the numbers at each pair of aligned vertices are different. Prove that the $ n\minus{}gons$ can be rotated to a position where at least two pairs of aligned vertices contain equal numbers.

We define the [i]repetition[/i] number of a positive integer $n$ to be the number of distinct digits of $n$ when written in base $10$. Prove that each positive integer has a multiple which has a repetition number less than or equal to $2$.

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?

[color=#000][b](i)[/b][/color] Let $a_1,a_2,...,a_n$ be n real numbers. Show that there exists some real number $\alpha$ such that $a_1+\alpha,a_2+\alpha,...,a_n+\alpha$ are all irrational. \\ [color=#000][b](ii)[/b][/color] Prove that such a satetement is not valid if all these are rquired to be rational. \\ [i]Hint (given in question) : Use Pigeon Hole Principle[/i]

Let $S$ be a set of $n$ elements and $S_1,\ S_2,\dots,\ S_k$ are subsets of $S$ ($k\geq2$), such that every one of them has at least $r$ elements. Show that there exists $i$ and $j$, with $1\leq{i}<j\leq{k}$, such that the number of common elements of $S_i$ and $S_j$ is greater or equal to: $r-\frac{nk}{4(k-1)}$