Circles $\omega$ and $\Omega$ meet at points $A$ and $B$. Let $M$ be the midpoint of the arc $AB$ of circle $\omega$ ($M$ lies inside $\Omega$). A chord $MP$ of circle $\omega$ intersects $\Omega$ at $Q$ ($Q$ lies inside $\omega$). Let $\ell_P$ be the tangent line to $\omega$ at $P$, and let $\ell_Q$ be the tangent line to $\Omega$ at $Q$. Prove that the circumcircle of the triangle formed by the lines $\ell_P$, $\ell_Q$ and $AB$ is tangent to $\Omega$. [i]Ilya Bogdanov, Russia and Medeubek Kungozhin, Kazakhstan[/i]

On the grid plane all possible broken lines with the following properties are constructed: each of them starts at the point $(0, 0)$, has all its vertices at integer points, each linear segment goes either up or to the right along the grid lines. For each such broken line consider the corresponding [i]worm[/i], the subset of the plane consisting of all the cells that share at least one point with the broken line. Prove that the number of worms that can be divided into dominoes (rectangles $2\times 1$ and $1\times 2$) in exactly $n > 2$ different ways, is equal to the number of positive integers that are less than n and relatively prime to $n$. (Ilke Chanakchi, Ralf Schiffler)

Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.

Prove that there exists an infinite number of relatively prime pairs $(m, n)$ of positive integers such that the equation \[x^3-nx+mn=0\] has three distint integer roots.

Given a right circular cone and a point $A$. Find the set of vertices of cones equal to the given one, with axes parallel to that of the given one, and with $A$ inside them. We shall assume that the cone is infinite in one side.

Let $ S$ be a set of $ 100$ points in the plane. The distance between every pair of points in $ S$ is different, with the largest distance being $ 30$. Let $ A$ be one of the points in $ S$, let $ B$ be the point in $ S$ farthest from $ A$, and let $ C$ be the point in $ S$ farthest from $ B$. Let $ d$ be the distance between $ B$ and $ C$ rounded to the nearest integer. What is the smallest possible value of $ d$?

Find the Green's function of the operator $d^2/dx^2-1$ and solve the equation \[\int_{-\infty}^{+\infty}e^{-|x-y|}u(y)dy=e^{-x^2}\]

Let $ABC$ be a triangle such that $AB \neq AC$. The internal bisector lines of the angles $ABC$ and $ACB$ meet the opposite sides of the triangle at points $B_0$ and $C_0$, respectively, and the circumcircle $ABC$ at points $B_1$ and $C_1$, respectively. Further, let $I$ be the incentre of the triangle $ABC$. Prove that the lines $B_0C_0$ and $B_1C_1$ meet at some point lying on the parallel through $I$ to the line $BC$. [i]Radu Gologan[/i]

The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L,$ without traveling along any portion of a road more than once. (Paula [i]is[/i] allowed to visit a city more than once.) How many different routes can Paula take? [asy] import olympiad; unitsize(50); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 4; ++j) { pair A = (j,i); dot(A); } } for (int i = 0; i < 3; ++i) { for (int j = 0; j < 4; ++j) { if (j != 3) { draw((j,i)--(j+1,i)); } if (i != 2) { draw((j,i)--(j,i+1)); } } } label("$A$", (0,2), W); label("$L$", (3,0), E); [/asy] $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3\qquad\textbf{(E) } 4$

The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.

In a triangle $ ABC $ that has perimeter $ P, $ prove that it's isosceles if and only if $$ P^2+\sin^2 (\angle ABC-\angle BCA) =4\cdot AB\cdot AC\cdot\cos^2\frac{\angle ABC}{2}\cdot\cos^2\frac{\angle BCA}{2} . $$

2. There are three hands on a clock. Each of them rotates in a normal direction at some non-zero speed, which can be wrong. In the morning the long and the short hands coincided. Just in three hours after that moment the long and the mid-length hands coincided. After next four hours the short and the mid-length hands coincided. Will it necessarily occur that all three hands will coincide? Alexandr Yuran

A set of points in the plane is called [i]sane[/i] if no three points are collinear and the angle between any three distinct points is a rational number of degrees. (a) Does there exist a countably infinite sane set $\mathcal{P}$? (b) Does there exist an uncountably infinite sane set $\mathcal{Q}$? [i]Proposed by Tony Wang[/i]

Let $f(x)$ be a polynomial of degree $3$ with real coefficients satisfying $|f(x)| = 12$ for $x = 1, 2, 3, 5, 6, 7$. Find $|f(0)|$.

Let $ a>0$, $ d>0$ and put $ f(x)\equal{}\frac{1}{a}\plus{}\frac{x}{a(a\plus{}d)}\plus{}\cdots\plus{}\frac{x^n}{a(a\plus{}d)\cdots(a\plus{}nd)}\plus{}\cdots$ Give a closed form for $ f(x)$.

Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points.