We consider all pairs (x, y) of real numbers such that $0\leq x \leq y \leq 1$.Let $M (x,y)$ the maximum value of the set $$A=\{xy, 1-x-y+xy, x+y-2xy\}.$$ Find the minimum value that $M(x,y)$ can take for all these pairs $(x,y)$.

Some computers of a computer room have a following network. Each computers are connected by three cable to three computers. Two arbitrary computers can exchange data directly or indirectly (through other computers). Now let's remove $K$ computers so that there are two computers, which can not exchange data, or there is one computer left. Let $k$ be the minimum value of $K$. Let's remove $L$ cable from original network so that there are two computers, which can not exchange data. Let $l$ be the minimum value of $L$. Show that $k=l$.

Let $n$ be a positive integer. An $n \times n$ matrix (a rectangular array of numbers with $n$ rows and $n$ columns) is said to be a platinum matrix if: [list=i] [*] the $n^2$ entries are integers from $1$ to $n$; [*] each row, each column, and the main diagonal (from the upper left corner to the lower right corner) contains each integer from $1$ to $n$ exactly once; and [*] there exists a collection of $n$ entries containing each of the numbers from $1$ to $n$, such that no two entries lie on the same row or column, and none of which lie on the main diagonal of the matrix. [/list] Determine all values of $n$ for which there exists an $n \times n$ platinum matrix.

Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that \[ \left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\] for every positive integer $n$.

Two rooks on a chessboard are said to be attacking each other if they are placed in the same row or column of the board. (a) There are eight rooks on a chessboard, none of them attacks any other. Prove that there is an even number of rooks on black fields. (b) How many ways can eight mutually non-attacking rooks be placed on the 9 £ 9 chessboard so that all eight rooks are on squares of the same color.

Determine the smallest natural number that has exactly $1994$ divisors.

In three cisterns of milk lies $780$ litres of milk. When we pour off from first cistern quarter of milk, from second cistern fifth of milk and from third cistern $\frac{3}{7}$ of milk, in all cisterns remain same amount of milk. How many milk is in cisterns?

Find the smallest nonnegative integer $n$ such that in every set of $n$ numbers there are always two distinct numbers such that their sum or difference is divisible by $2022$.

Find all real solutions of the system \[\begin{cases}x^2 + y^2 - z(x + y) = 2, \\ y^2 + z^2 - x(y + z) = 4, \\ z^2 + x^2 - y(z + x) = 8.\end{cases}\]

Let $ p$ be a prime number, $ p\not \equal{} 3$, and integers $ a,b$ such that $p\mid a+b$ and $ p^2\mid a^3 \plus{} b^3$. Prove that $ p^2\mid a \plus{} b$ or $ p^3\mid a^3 \plus{} b^3$.

The incircle of a triangle $ABC$ is tangent to $BC$ at $D$. Let $H$ and $\Gamma$ denote the orthocenter and circumcircle of $\triangle ABC$. The $B$-mixtilinear incircle, centered at $O_B$, is tangent to lines $BA$ and $BC$ and internally tangent to $\Gamma$. The $C$-mixtilinear incircle, centered at $O_C$, is defined similarly. Suppose that $\overline{DH} \perp \overline{O_BO_C}$, $AB = \sqrt3$ and $AC = 2$. Find $BC$.

Let $I$ be the incenter of the triangle $ABC$ ($AB < BC$). Let $M$ be the midpoint of $AC$, and let $N$ be the midpoint of the arc $AC$ of the circumcircle of $ABC$ which contains $B$. Prove that $\angle IMA = \angle INB$.

What is the greatest number of consecutive integers whose sum is $45 ?$ $\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$

Regular hexagon $NOSAME$ with side length $1$ and square $UDON$ are drawn in the plane such that $UDON$ lies outside of $NOSAME$. Compute $[SAND] + [SEND]$, the sum of the areas of quadrilaterals $SAND$ and $SEND$.

In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.

Show that no matter how $15$ points are plotted within a circle of radius $2$ (circle border included), there will be a circle with radius $1$ (circle border including) which contains at least three of the $15$ points.

Let $a,b,c\ge 0$ be three real numbers such that $$ab+bc+ca+2abc=1.$$ Prove that $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 2$ and determine equality cases.

Find $P(x)\in Z[x]$ st : $P(n)|2557^{n}+213.2014$ with any $n\in N^{*}$

Let $a$ and $b$ be distinct positive integers, bigger that $10^6$, such that $(a+b)^3$ is divisible with $ab$. Prove that $ \mid a-b \mid > 10^4$

Let $\mathcal{H}$ be the set of all lines in the plane. Call a function $f:\mathbb{R}^2\to\mathcal{H}$ [i]polarising[/i], if $P\in f(Q)$ implies $Q\in f(P)$ for any pair of points $P,Q\in\mathbb{R}^2.$ [list=a] [*]Show that there is no surjective polarising function. [*]Give an example of an injective polarising function. [*]Prove that for every injective polarising function there exists a point $P$ in the plane for which $P\in f(P).$ [/list]

Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$. Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.

How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$ $\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$

A contractor has two teams of workers: team $A$ and team $B$. Team $A$ can complete a job in $12$ days and team $B$ can do the same job in $36$ days. Team $A$ starts working on the job and team $B$ joins team $A$ after four days. The team $A$ withdraws after two more days. For how many more days should team $B$ work to complete the job?

Let $x_0=a,x_1=b$ and $x_{n+1}=2x_n-9x_{n-1}$ for each $n\in\mathbb N$, where $a,b$ are integers. Find the necessary and sufficient condition on $a$ and $b$ for the existence of an $x_n$ which is a multiple of $7$.

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude from $A$. The circle with centre $A$ passing through $D$ intersects the circumcircle of triangle $ABC$ in $X$ and $Y$ , in such a way that the order of the points on this circumcircle is: $A, X, B, C, Y$ . Show that $\angle BXD = \angle CYD$.