Find all pairs of positive integers $ (a, b)$ such that \[ ab = gcd(a, b) + lcm(a, b). \]

Prove that the circle with radius $2$ can be completely covered with $7$ unit circles

Let there be $ n \ge 2$ real numbers such that none of them is greater than the arithmetic mean of the other numbers. Prove that all the numbers are equal.

Triangle $ ABC$ is isosceles with $ AC = BC$ and $ \angle{C} = 120^o$. Points $ D$ and $ E$ are chosen on segment $ AB$ so that $ |AD| = |DE| = |EB|$. Find the sizes of the angles of triangle $ CDE$.

Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a ship a figure made up of unit squares connected by common edges. We call a fleet a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a fleet to which no new ship can be added.

Calculate the sum $$\frac{1}{1+2^{-2006}}+...+ \frac{1}{1+2^{-1}}+ \frac{1}{1+2^{0}}+ \frac{1}{1+2^{1}}+...+ \frac{1}{1+2^{2006}}$$

Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares. a) Give an example of such numbers $a, b$ and $c$. b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$

Let $AG, CH$ be the angle bisectors of a triangle $ABC$. It is known that one of the intersections of the circles of triangles $ABG$ and $ACH$ lies on the side $BC$. Prove that the angle $BAC$ is $60 ^o$

Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $

Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a ship a figure made up of unit squares connected by common edges. We call a fleet a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, for each its representation as a sum of positive integers, there exists a fleet such that the summands are exactly the numbers of squares contained in individual ships.

Find the greatest possible value of $ sin(cos x) + cos(sin x)$ and determine all real numbers x, for which this value is achieved.

In a right triangle, the length of one side is a prime and the lengths of the other side and the hypotenuse are integral. The ratio of the triangle perimeter and the incircle diameter is also an integer. Find all possible side lengths of the triangle.

The sequence $ (F_n)$ of Fibonacci numbers satisfies $ F_1 = 1, F_2 = 1$ and $ F_n = F_{n-1} +F_{n-2}$ for all $ n \ge 3$. Find all pairs of positive integers $ (m, n)$, such that $ F_m . F_n = mn$.

In a triangle ABC with circumcentre O and centroid M, lines OM and AM are perpendicular. Let AM intersect the circumcircle of ABC again at A′. Let lines BA′ and AC intersect at D and let lines CA′ and AB intersect at E. Prove that the circumcentre of triangle ADE lies on the circumcircle of ABC.

A pawn is placed on a square of a $ n \times n$ board. There are two types of legal moves: (a) the pawn can be moved to a neighbouring square, which shares a common side with the current square; or (b) the pawn can be moved to a neighbouring square, which shares a common vertex, but not a common side with the current square. Any two consecutive moves must be of different type. Find all integers $ n \ge 2$, for which it is possible to choose an initial square and a sequence of moves such that the pawn visits each square exactly once (it is not required that the pawn returns to the initial square).

We call a ship a figure made up of unit squares connected by common edges. Prove that if there is an odd number of possible different ships consisting of n unit squares on a $ 10 \times 10$ board, then n is divisible by 4.

Find the smallest possible distance of points $ P$ and $ Q$ on a $ xy$-plane, if $ P$ lies on the line $ y = x$ and $ Q$ lies on the curve $ y = 2^x$.

Prove or disprove the following statements. a) For every integer $ n \ge 3$, there exist $ n$ pairwise distinct positive integers such that the product of any two of them is divisible by the sum of the remaining $ n - 2$ numbers. b) For some integer $ n \ge 3$, there exist $ n$ pairwise distinct positive integers, such that the sum of any $ n - 2$ of them is divisible by the product of the remaining two numbers.

Let O be the circumcentre of an acute triangle ABC and let A′, B′ and C′ be the circumcentres of triangles BCO, CAO and ABO, respectively. Prove that the area of triangle ABC does not exceed the area of triangle A′B′C′.

The Ababi alphabet consists of letters A and B, and the words in the Ababi language are precisely those that can be formed by the following two rules: 1) A is a word. 2) If s is a word, then $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ denotes a word that is obtained by replacing all letters A in s with letters B, and vice versa; and $ x \oplus y$ denotes the concatenation of x and y. The Ululu alphabet consists also of letters A and B and the words in the Ululu language are precisely those that can be formed by the following two rules: 1) A is a word. 2) If s is a word, $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ is defined as above and $ x \oplus y$ is a word obtained from words x and y of equal length by writing the letters of x and y alternatingly, starting from the first letter of x. Prove that the two languages consist of the same words.