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.
There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.
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?
9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.
Is it possible to arrange 1000 line segments in a plane so that both ends of each line segment rest strictly inside another line segment?
Some open sectors – that is sectors of circles with infinite radii – completely cover a plane. Prove that the sum of the angles of these sectors is no less than \(360^\circ\).
A spherical planet is surrounded by 25 point asteroids. Prove, that at any given moment there will be a point on the surface of the planet from which an astronomer will not be able to see more than 11 asteroids.
On a line, there are 50 segments. Prove that either it is possible to find some 8 segments all of which have a shared intersection, or there can be found 8 segments, no two of which intersect.
10 people collected a total of 46 mushrooms in a forest. It is known that no two people collected the same number of mushrooms. How many mushrooms did each person collect?
10 magazines lie on a coffee table, completely covering it. Prove that you can remove five of them so that the remaining magazines will cover at least half of the table.