Find the largest integer such that every number after the first is one less than the previous one and is divisible by each of its own numbers.

Two medians drawn from vertices A and B of triangle ABC are perpendicular. Prove that side AB is the shortest side of ABC.

The headteacher wants to hire a certain number of new teachers in addition to existing teachers. If he hired an additional $10$ teachers, the number of school students would be reduced number per teacher by $5$. However, if the headmaster hired $20$ new teachers, the number of students per teacher would be reduced by $8$. How many students and how many there are teachers in this school?

The figure shows a figure of $5$ unit squares, a Greek cross. What is the largest number of Greek crosses that can be placed on a grid of dimensions $8 \times 8$ without any overlaps, with each unit square covering just one square in a grid?

Juhan wants to order by weight five balls of pairwise different weight, using only a balance scale. First, he labels the balls with numbers 1 to 5 and creates a list of weighings, such that each element in the list is a pair of two balls. Then, for every pair in the list, he weighs the two balls against each other. Can Juhan sort the balls by weight, using a list with less than 10 pairs?

The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers. a) Find all possibilities for the three numbers that are not included in this number. b) Give an example of such a number.

Two radii OA and OB of a circle c with midpoint O are perpendicular. Another circle touches c in point Q and the radii in points C and D, respectively. Determine $ \angle{AQC}$.

Prove that the sum of the squares of any three pairwise different positive odd integers can be represented as the sum of the squares of six (not necessarily different) positive integers.

Two triangles are drawn on a plane in such a way that the area covered by their union is an n-gon (not necessarily convex). Find all possible values of the number of vertices n.

The identifier of a book is an n-tuple of numbers 0, 1, .... , 9, followed by a checksum. The checksum is computed by a fixed rule that satisfies the following property: whenever one increases a single number in the n-tuple (without modifying the other numbers), the checksum also increases. Find the smallest possible number of required checksums if all possible n-tuples are in use.

Find all real numbers a such that all solutions to the quadratic equation $ x^2 - ax + a = 0$ are integers.

A 3-dimensional chess board consists of $ 4 \times 4 \times 4$ unit cubes. A rook can step from any unit cube K to any other unit cube that has a common face with K. A bishop can step from any unit cube K to any other unit cube that has a common edge with K, but does not have a common face. One move of both a rook and a bishop consists of an arbitrary positive number of consecutive steps in the same direction. Find the average number of possible moves for either piece, where the average is taken over all possible starting cubes K.

A circle passing through the endpoints of the leg AB of an isosceles triangle ABC intersects the base BC in point P. A line tangent to the circle in point B intersects the circumcircle of ABC in point Q. Prove that P lies on line AQ if and only if AQ and BC are perpendicular.

Find all pairs $ (m, n)$ of positive integers such that $ m^n - n^m = 3$.

Some circles of radius 2 are drawn on the plane. Prove that the numerical value of the total area covered by these circles is at least as big as the total length of arcs bounding the area.

Consider a cylinder and a cone with a common base such that the volume of the part of the cylinder enclosed in the cone equals the volume of the part of the cylinder outside the cone. Find the ratio of the height of the cone to the height of the cylinder.

Let $ x, y, z$ be positive real numbers such that $ x^n, y^n$ and $ z^n$ are side lengths of some triangle for all positive integers $ n$. Prove that at least two of x, y and z are equal.

Does there exist an equilateral triangle (a) on a plane; (b) in a 3-dimensional space; such that all its three vertices have integral coordinates?

Let $a, b,c$ be positive integers such that $gcd(a, b, c) = 1$ and each product of two is divided by the third. a) Prove that each of these numbers is equal to the least two remaining numbers the quotient of the coefficient and the highest coefficient. b) Give an example of one of these larger numbers $a, b$ and $c$

In a grid of dimensions $n \times n$, a part of the squares is marked with crosses such that in each at least half of the $4 \times 4$ squares are marked. Find the least possible the total number of marked squares in the grid.