101 points are marked on a plane; not all of the points lie on the same straight line. A red pencil is used to draw a straight line passing through each possible pair of points. Prove that there will always be a marked point on the plane through which at least 11 red lines pass.
33 representatives of four different races – humans, elves, gnomes, and goblins – sit around a round table.
It is known that humans do not sit next to goblins, and that elves do not sit next to gnomes. Prove that some two representatives of the same peoples must be sitting next to one another.
Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?
A number is written on each edge of a cube. The sum of the 4 numbers on the adjacent edges is written on each face. Place the numbers \(1\) and \(-1\) on the edges so that the numbers written on the faces are all different.
A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?
15 MPs take part in a debate. During the debate, each one criticises exactly \(k\) of the 14 other contributors. For what minimum value of \(k\) is it possible to definitively state that there will be two MPs who have criticised one another?
In the number \(1234096\dots\) each digit, starting with the 5th digit is equal to the final digit of the sum of the previous 4 digits. Will the digits 8123 ever occur in that order in a row in this number?
7 natural numbers are written around the edges of a circle. It is known that in each pair of adjacent numbers one is divisible by the other. Prove that there will be another pair of numbers that are not adjacent that share this property.
Is it possible to transport 50 stone blocks, whose masses are equal to \(370, 372,\dots, 468\) kg, from a quarry on seven 3-tonne trucks?
The city plan is a rectangle of \(5 \times 10\) cells. On the streets, a one-way traffic system is introduced: it is allowed to go only to the right and upwards. How many different routes lead from the bottom left corner to the upper right?