All the streets in a city run in one of two perpendicular directions, forming unit squares. Organizers of a car race want to mark down a closed race track in the city in such a way that it would not go through any of the crossings twice and that the track would turn 90◦ right or left at every crossing. Find all possible values of the length of the track.

After the schoolday is over, Juku must attend an extra math class. The teacher writes a quadratic equation $ x^2+ p_1x+q_1 = 0$ with integer coefficients on the blackboard and Juku has to find its solutions. If they are not both integers, Jukumay go home. If the solutions are integers, then the teacher writes a new equation $ x^2 + p_2x + q_2 = 0,$ where $ p_2$ and $ q_2$ are the solutions of the previous equation taken in some order, and everything starts all over. Find all possible values for $ p_1$ and $ q_1$ such that the teacher can hold Juku at school forever.

Let $ ABC$ be an acute triangle and choose points $ A_1, B_1$ and $ C_1$ on sides $ BC, CA$ and $ AB$, respectively. Prove that if the quadrilaterals $ ABA_1B_1, BCB_1C_1$ and $ CAC_1A_1$ are cyclic then their circumcentres lie on the sides of $ ABC$.

Martin invented the following algorithm. Let two irreducible fractions $ \frac{s_1}{t_1}$ and $ \frac{s_2}{t_2}$ be given as inputs, with the numerators and denominators being positive integers. Divide $ s_1$ and $ s_2$ by their greatest common divisor $ c$ and obtain $ a_1$ and $ a_2$, respectively. Similarily, divide $ t_1$ and $ t_2$ by their greatest common divisor $ d$ and obtain $ b_1$ and $ b_2$, respectively. After that, form a new fraction $ \frac{a_1b_2 + a_2b_1}{t_1b_2}$, reduce it, and multiply the numerator of the result by $ c$. Martin claims that this algorithm always finds the sum of the original fractions as an irreducible fraction. Is his claim correct?

Two players A and B play the following game. Initially, there are $ m$ equal positive integers $ n$ written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number k is the smallest among all positive integers on the board, the player replaces it with $ k-1$; if not, the player replaces it with the smallest positive number on the board. The player who first turns all the numbers into zeroes, wins. Who wins if both players use their best strategies?

Kati cut two equal regular $ n-gons$ out of paper. To the vertices of both $ n-gons$, she wrote the numbers 1 to $ n$ in some order. Then she stabbed a needle through the centres of these $ n-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-gons$ can be rotated to a position where at least two pairs of aligned vertices contain equal numbers.

A real-valued function $ f$ satisfies for all reals $ x$ and $ y$ the equality \[ f (xy) = f (x)y + x f (y). \] Prove that this function satisfies for all reals $ x$ and $ y \ne 0$ the equality \[ f\left(\frac{x}{y}\right)=\frac{f (x)y - x f (y)}{y^2} \]

Four points $ A, B, C, D$ are chosen on a circle in such a way that arcs $ AB, BC,$ and $ CD$ are of the same length and the $ arc DA$ is longer than these three. Line $ AD$ and the line tangent to the circle at $ B$ intersect at $ E$. Let $ F$ be the other endpoint of the diameter starting at $ C$ of the circle. Prove that triangle $ DEF$ is equilateral.

In the sequence $ (a_n)$ with general term $ a_n = n^3 - (2n + 1)^2$, does there exist a term that is divisible by 2006?

Let $ n \ge 2$ be a fixed integer and let $ a_{i,j} (1 \le i < j \le n)$ be some positive integers. For a sequence $ x_1, ... , x_n$ of reals, let $ K(x_1, .... , x_n)$ be the product of all expressions $ (x_i - x_j)^{a_{i,j}}$ where $ 1 \le i < j \le n$. Prove that if the inequality $ K(x_1, .... , x_n) \ge 0$ holds independently of the choice of the sequence $ x_1, ... , x_n$ then all integers $ a_{i,j}$ are even.