Two players, A and B, play the following game: they retire coins of a pile which contains initially 2006 coins. The players play removing alternatingly, in each move, from 1 to 7 coins, each player keeps the coins that retires. If a player wishes he can pass(he doesn't retire any coin), but to do that he must pay 7 coins from the ones he retired from the pile in past moves. These 7 coins are taken to a separated box and don't interfere in the game any more. The winner is the one who retires the last coin, and A starts the game. Determine which player can win for sure, it doesn't matter how the other one plays. Show the winning strategy and explain why it works.

Let $S$ be the solid in three-dimensional space consisting of all points $(x,y,z)$ satisfying the following six simultaneous conditions: $$ x,y,z \geq 0, \;\; x+y+z\leq 11, \;\; 2x+4y+3z \leq 36, \;\; 2x+3z \leq 44.$$ a) Determine the number $V$ of vertices of $S.$ b) Determine the number $E$ of edges of $S.$ c) Sketch in the $bc$-plane the set of points $(b, c)$ such that $(2,5,4)$ is one of the points $(x, y, z)$ at which the linear function $bx + cy + z$ assumes its maximum value on $S.$

The inscribed circle of the non-isosceles triangle $ABC$ touches sides $AB, BC$ and $AC$ at points $C_1, A_1$ and $B_1$, respectively. The circumscribed circle of the triangle $A_1BC_1$ intersects the lines $B_1A_1$ and $B_1C_1$ at the points $A_0$ and $C_0$, respectively. Prove that the orthocenter of triangle $A_0BC_0$, the center of the inscribed circle of triangle $ABC$ and the midpoint of the $AC$ lie on one straight line.

Let $a,b,c$ be the side lengths of any triangle. Prove that $$\frac{a}{\sqrt{2b^2+2c^2-a^2}}+\frac{b}{\sqrt{2c^2+2a^2-b^2 }}+\frac{c}{\sqrt{2a^2+2b^2-c^2}}\ge \sqrt{3}.$$ [i](Zhuge Liang)[/i]

Let $0 < u < 1$ and define \[u_1 = 1 + u, \quad u_2 = \frac{1}{u_1} + u, \quad \dots, \quad u_{n + 1} = \frac{1}{u_n} + u, \quad n \ge 1.\] Show that $u_n > 1$ for all values of $n = 1$, 2, 3, $\dots$.

The set of three-digit natural numbers formed from digits $1,2, 3, 4, 5, 6$ is called [i]nice [/i] if it satisfies the following condition: for any two different digits from $1, 2, 3, 4, 5, 6$ there exists a number from the set which contains both of them. For any nice set we calculate the sum of all its elements. Determine the smallest possible value of these sums. (E. Barabanov)

The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.

Evaluate $\int_{0}^{\pi/2}\frac{dx}{\left(\sqrt{\sin x}+\sqrt{\cos x}\right)^4}$.

a) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31$ (the product of primes $2$ to $31$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers. Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$. b) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37$ (the product of primes $2$ to $37$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers. Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$.

On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue. At the intersection of two walls there are four corners: two diagonally opposite corners where a red side and a blue side meet, one corner where two red sides meet, and one corner where two blue sides meet. At each such intersection, there is a two-way door connecting the two diagonally opposite corners at which sides of different colours meet. After Merlin paints the walls, Morgana then places some knights in the labyrinth. The knights can walk through doors, but cannot walk through walls. Let $k(\mathfrak{L})$ be the largest number $k$ such that, no matter how Merlin paints the labyrinth $\mathfrak{L},$ Morgana can always place at least $k$ knights such that no two of them can ever meet. For each $n,$ what are all possible values for $k(\mathfrak{L}),$ where $\mathfrak{L}$ is a labyrinth with $n$ walls?

For integers $z$, let $\#(z)$ denote the number of integer ordered pairs $(x, y)$ that satisfy $x^2 - xy + y^2 = z$. How many integers $z$ between $0$ and $150$ inclusive satisfy $\#(z) \equiv 6$ (mod $12$)?

Prove that every multiple of $3$ can be written as a sum of four cubes (positive or negatives).

For each positive integer $n$ let $a_n$ be the least positive integer multiple of $23$ such that $a_n\equiv1\pmod{2^n}$. Find the number of positive integers $n$ less than or equal to $1000$ that satisfy $a_n=a_{n+1}$.

Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\overline{AB}$ and $\overline{AC}$, respectively, such that $AP=AQ$. Let $S$ and $R$ be distinct points on segment $\overline{BC}$ such that $S$ lies between $B$ and $R$, $\angle BPS=\angle PRS$, and $\angle CQR=\angle QSR$. Prove that $P,Q,R,S$ are concyclic (in other words, these four points lie on a circle).

A ticket for the tram costs 1 leva. On the queue in front of the ticket seller are standing $n$ people with a banknote of 1 leva and $m$ people with a banknote of 2 leva. The ticket seller has no money in his cash deck so he can only sell a ticket to a buyer with a banknote of 2 leva, if he has at least 1 banknote of 1 leva. Determine the probability that the ticket seller could sell tickets to all of the people standing in the queue.

Which of the polynomials, $(1+x^2 -x^3)^{1000}$ or $(1-x^2 +x^3)^{1000}$, has the greater coefficient of $x^{20}$ after expansion and collecting the terms?

Suppose $m$ and $n$ are positive integers such that $m^2+n^2+m$ is divisible by $mn$. Prove that $m$ is a square number.

The city of Atlantis is built on an island represented by $[ -1, 1]$, with skyline initially given by $f(x) = 1 - |x| $. The sea level is currently $y=0$, but due to global warming, it is rising at a rate of $0.01$ a year. For any position $-1 < x < 1$, while the building at $x$ is not completely submerged, then it is instantaneously being built upward at a rate of $r$ per year, where $r$ is the distance (along the $x$-axis) from this building to the nearest completely submerged building. How long will it be until Atlantis becomes completely submerged? [i]Proposed by Ethan Tan[/i]

Given an acute and scalene triangle $ABC$ with $AB<AC$ and random line $(e)$ that passes throuh the center of the circumscribed circles $c(O,R)$. Line $(e)$, intersects sides $BC,AC,AB$ at points $A_1,B_1,C_1$ respectively (point $C_1$ lies on the extension of $AB$ towards $B$). Perpendicular from $A$ on line $(e)$ and $AA_1$ intersect circumscribed circle $c(O,R)$ at points $M$ and $A_2$ respectively. Prove that a) points $O,A_1,A_2, M$ are consyclic b) if $(c_2)$ is the circumcircle of triangle $(OBC_1)$ and $(c_3)$ is the circumcircle of triangle $(OCB_1)$, then circles $(c_1),(c_2)$ and $(c_3)$ have a common chord

Show that amongst any $ 8$ points in the interior of a $7 \times 12$ rectangle, there exists a pair whose distance is less than $5$. Note: The interior of a rectangle excludes points lying on the sides of the rectangle.

Let $\{A_n | n = 1, 2, \cdots \} $ be a set of points in the plane such that for each $n$, the disk with center $A_n$ and radius $2^n$ contains no other point $A_j$ . For any given positive real numbers $a < b$ and $R$, show that there is a subset $G$ of the plane satisfying: [b](i)[/b] the area of $G$ is greater than or equal to $R$; [b](ii) [/b]for each point $P$ in $G$, $a < \sum_{n=1}^{\infty} \frac{1}{|A_nP|} <b.$

Let $a,b,c$ be distinct real numbers and $P(x)$ a polynomial with real coefficients. Suppose that the remainders of $P(x)$ upon division by $(x-a), (x-b)$ and $(x-c)$ are $a,b$ and $c$, respectively. Find the polynomial that is obtained as the remainder of $P(x)$ upon division by $(x-a)(x-b)(x-c)$.

Let $A',\,B',\,C'$ be points, in which excircles touch corresponding sides of triangle $ABC$. Circumcircles of triangles $A'B'C,\,AB'C',\,A'BC'$ intersect a circumcircle of $ABC$ in points $C_1\ne C,\,A_1\ne A,\,B_1\ne B$ respectively. Prove that a triangle $A_1B_1C_1$ is similar to a triangle, formed by points, in which incircle of $ABC$ touches its sides.

Bas has coloured each of the positive integers. He had several colours at his disposal. His colouring satises the following requirements: • each odd integer is coloured blue, • each integer $n$ has the same colour as $4n$, • each integer $n$ has the same colour as at least one of the integers $n+2$ and $n + 4$. Prove that Bas has coloured all integers blue.

Triangle $ OAB$ has $ O \equal{} (0,0)$, $ B \equal{} (5,0)$, and $ A$ in the first quadrant. In addition, $ \angle{ABO} \equal{} 90^\circ$ and $ \angle{AOB} \equal{} 30^\circ$. Suppose that $ \overline{OA}$ is rotated $ 90^\circ$ counterclockwise about $ O$. What are the coordinates of the image of $ A$? $ \textbf{(A)}\ \left( \minus{} \frac {10}{3}\sqrt {3},5\right) \qquad \textbf{(B)}\ \left( \minus{} \frac {5}{3}\sqrt {3},5\right) \qquad \textbf{(C)}\ \left(\sqrt {3},5\right) \qquad \textbf{(D)}\ \left(\frac {5}{3}\sqrt {3},5\right) \\ \textbf{(E)}\ \left(\frac {10}{3}\sqrt {3},5\right)$