Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}$. Let $O$ be the circumcenter of the triangle $ABC$ and let $\Gamma$ be the circumcircle of the triangle $BOC$. Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C$. Let $ON$ be a diameter of the circle $\Gamma$. Prove that the quadrilateral $APNQ$ is a parallelogram.
Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?
Find all functions $f$ from the set $\{\mathbf{R}\}$ of real numbers into $\{\mathbf{R}\}$ which satisfy for all $x, y, z \in \{\mathbf{R}\}$ the identity
$$
f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2 x y+f(z))+2 f(x z-y z)
$$
Let $ABC$ be an acute triangle satisfying the condition $AB \gt BC$ and $AC \gt BC$. Denote by $O$ and $H$ the circumcenter and the orthocenter, respectively, of the triangle $ABC$. Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and that the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A$. Prove that the circumcenter of the triangle $MNH$ lies on the line $OH$.
For a positive integer $k$, call an integer a pure $k$-th power if it can be represented as $m^{k}$ for some integer $m$. Show that for every positive integer $n$ there exist $n$ distinct positive integers such that their sum is a pure $2009$-th power, and their product is a pure $2010$-th power.
Let $n$ be a positive integer. We call *smooth* a sequence of integers $a_1, a_2, \dots, a_k$, with $1 \le a_i \le n$, if there exists an integer $m$, with $1 \le m \lt k$, such that $a_1 = a_{k-m+1}, a_2 = a_{k-m+2}, \dots, a_m = a_k$. Furthermore a sequence is *universal* if each of the sequences obtained through replacing $a_k$ by $1, 2, \dots, n$ is smooth. For each $n$ find a universal sequence of minimum length.
In a school with 5 grades there are 250 girls and 250 boys. Each grade has 100 students. Teams of one girl and one boy from the same grade must be formed for a contest. At least 19 students in each grade are girls and at least 19 are boys. Find the greatest number of teams that can be formed with certainty.
Players $A$ and $B$ play a game as follows. Initially $A$ arranges the numbers $1, 2, \dots, n$ in a row as he wishes; $n$ is a given positive integer. Next, $B$ chooses one number and puts a stone on it. Then $A$ moves the stone to an adjacent number, $B$ does the same and so on. The stone can be placed on number $k$ at most $k$ times, $k = 1, \dots, n$; the initial move of $B$ is counted. The one who cannot move loses. For each $n$ determine who has a winning strategy.
The numbers $1, 2, \dots, 2010$ are written in a row. Two players take turns in writing $+$ or $\times$ between two consecutive numbers for as long as this is possible. The first player wins if the obtained algebraic sum is divisible by $3$; otherwise the second player wins. Find a winning strategy for one of the players.
In triangle $ABC$ point $P$ divides side $AB$ in ratio $\frac{AP}{PB} = \frac{1}{4}$. The perpendicular bisector of segment $PB$ intersects side $BC$ at point $Q$. If $\text{area}(PQC) = \frac{4}{25}\text{area}(ABC)$ and $AC = 7$, find $BC$.
Given several integers, it is allowed to replace two of them by their nonnegative difference. The operation is repeated until only one number remains. If the initial numbers are $1, 2, \ldots, 2010$, what can be the last number remaining?
In the parallelogram $ABCD$ point $G$ is chosen on side $AB$. Consider the circle through $A$ and $G$ that is tangent to the extension of $CB$ beyond $B$ at point $P$. The extension of $DG$ beyond $G$ intersects the circle at $L$. If the quadrilateral $GLBC$ is cyclic, prove that $AB = PC$.
There are given $2k$ boxes ($k \ge 2$) with $2k-1$ pebbles in each one. A legal move is to choose $2k-2$ boxes and remove one pebble from each one of them. Players $A$ and $B$ make moves alternately; $A$ goes first. A player wins if a move of his empties two boxes. Determine which player has a winning strategy.
On an infinite sheet of grid paper 999 grid squares are colored black. Call *special* a rectangle with sides on grid lines if it has two black opposite corner cells (rectangles with side 1 are included). Let $N$ be the maximum number of black cells in a special rectangle for a given configuration. Find the minimum of $N$ over all configurations.
In the acute-angled triangle $ABP$ ($AB \gt BP$) the altitudes are $BH$, $PQ$ and $AS$. The extension of $QS$ intersects line $AP$ at $C$. The extension of $HS$ intersects $BC$ at $L$. If $HS = SL$ and $HL$ is perpendicular to $BC$, compute $\frac{SL}{SC}$.
1000 balls of mass $0.38$ and 5000 balls of mass $0.038$ must be packed in boxes. A box can contain any collection of balls with total mass at most $1$. Find the minimum number of boxes needed.
a) Several distinct positive integers have the property that the sum of every three of them is a prime number. At most how many of them are there?
b) Several distinct integers (not necessarily positive) have the property that the sum of every three of them is positive and also a prime number. At most how many of them are there?