Let $A_1, A_2, ..., A_m$ and $B_2 , B_3,..., B_n$ be the points on a circle such that $A_1A_2... A_n$ is a regular $m$-gon and $A_1B_2...B_n$ is a regular $n$-gon whereby $n > m$ and the point $B_2$ lies between $A_1$ and $A_2$. Find $\angle B_2A_1A_2$.

Find all positive integers $n$ such that $n+ \left[ \frac{n}{6} \right] \ne \left[ \frac{n}{2} \right] + \left[ \frac{2n}{3} \right]$

In the rectangle $ABCD$ with $|AB|<2 |AD|$, let $E$ be the midpoint of $AB$ and $F$ a point on the chord $CE$ such that $\angle CFD = 90^o$. Prove that $FAD$ is an isosceles triangle.

Ella the Witch was mixing a magic elixir which consisted of three components: $140$ ml of reindeer moss tea, $160$ ml of fly agaric extract, and $50$ ml of moonshine. She took an empty $350$ ml bottle, poured $140$ ml of reindeer moss tea into it and started adding fly agaric extract when she was disturbed by its black cat Mehsto. So she mistakenly poured too much fly agaric extract into the bottle and noticed her fault only later when the bottle Riled before all $50$ ml of moonshine was added. Ella made quick calculations, carefully shaked up the contents of the bottle, poured out some part of liquid and added some amount of mixture of reindeer moss tea and fly agaric extract taken in a certain proportion until the bottle was full again and the elixir had exactly the right compositsion. Which was the proportion of reindeer moss tea and fly agaric extract in the mixture that Ella added into the bottle?

Is it possible to cover an $n \times n$ chessboard which has its center square cut out with tiles shown in the picture (each tile covers exactly $4$ squares, tiles can be rotated and turned around) if a) $n = 5$, b) $n = 2003$?

The picture shows $10$ equal regular pentagons where each two neighbouring pentagons have a common side. The smaller circle is tangent to one side of each pentagon and the larger circle passes through the opposite vertices of these sides. Find the area of the larger circle if the area of the smaller circle is $1$.

Find all possible integer values of $\frac{m^2+n^2}{mn}$ where m and n are integers.

In the acute-angled triangle $ABC$ all angles are greater than $45^o$. Let $AM$ and $BN$ be the heights of this triangle and let $X$ and $Y$ be the points on $MA$ and $NB$, respecively, such that $|MX| =|MB|$ and $|NY| =|NA|$. Prove that $MN$ and $XY$ are parallel.

Let $a, b$, and $c$ be positive real numbers not greater than $2$. Prove the inequality $\frac{abc}{a + b + c} \le \frac43$

The game Clobber is played by two on a strip of $2k$ squares. At the beginning there is a piece on each square, the pieces of both players stand alternatingly. At each move the player shifts one of his pieces to the neighbouring square that holds a piece of his opponent and removes his opponent’s piece from the table. The moves are made in turn, the player whose opponent cannot move anymore is the winner. Prove that if for some $k$ the player who does not start the game has the winning strategy, then for $k + 1$ and $k + 2$ the player who makes the first move has the winning strategy.

Juhan is touring in Europe. He stands on a highway and watches cars. There are three cars driving along the highway at constant speeds: an Opel and a Trabant in one direction and a Mercedes in the opposite direction. At the moment when the Trabant passes Juhan, the Opel and the Mercedes lie at equal distances from him in opposite directions. At the moment when the Mercedes passes Juhan, the Opel and the Trabant lie at equal distances from him in opposite directions. Prove that at the moment when the Opel passes Juhan, also the Mercedes and the Trabant lie at equal distances from him in opposite directions.

Prove that for all positive real numbers $a, b$, and $c$ , $\sqrt[3]{abc}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 2\sqrt3$. When does the equality occur?

Let $ABC$ be a triangle and $A_1, B_1, C_1$ points on $BC, CA, AB$, respectively, such that the lines $AA_1, BB_1, CC_1$ meet at a single point. It is known that $A, B_1, A_1, B$ are concyclic and $B, C_1, B_1, C$ are concyclic. Prove that a) $C, A_1, C_1, A$ are concyclic, b) $AA_1,, BB_1, CC_1$ are the heights of $ABC$.

Prove that there exist infinitely many positive integers $n$ such that $\sqrt{n}$ is not an integer and $n$ is divisible by $[\sqrt{n}] $.

For which positive integers $n$ is it possible to cover a $(2n+1) \times (2n+1)$ chessboard which has one of its corner squares cut out with tiles shown in the figure (each tile covers exactly $4$ squares, tiles can be rotated and turned around)?

Jiiri and Mari both wish to tile an $n \times n$ chessboard with cards shown in the picture (each card covers exactly one square). Jiiri wants that for each two cards that have a common edge, the neighbouring parts are of different color, and Mari wants that the neighbouring parts are always of the same color. How many possibilities does Jiiri have to tile the chessboard and how many possibilities does Mari have?

Solve the equation $\sqrt{x} = \log_2 x$.

Let $ABC$ be a triangle with $\angle C = 90^o$ and $D$ a point on the ray $CB$ such that $|AC| \cdot |CD| = |BC|^2$. A parallel line to $AB$ through $D$ intersects the ray $CA$ at $E$. Find $\angle BEC$.

Call a positive integer lonely if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that a) all primes are lonely, b) there exist infinitely many non-lonely positive integers.

On a lottery ticket a player has to mark $6$ numbers from $36$. Then $6$ numbers from these $36$ are drawn randomly and the ticket wins if none of the numbers that came out is marked on the ticket. Prove that a) it is possible to mark the numbers on $9$ tickets so that one of these tickets always wins, b) it is not possible to mark the numbers on $8$ tickets so that one of tickets always wins.