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.
It is known that in a convex \(n\)-gon (\(n > 3\)) no three diagonals pass through one point. Find the number of points (other than the vertex) where pairs of diagonals intersect.
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.
It is known that \[35! = 10333147966386144929 * 66651337523200000000.\] Find the number replaced by an asterisk.
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?
There are 30 students in the class. Prove that the probability that some two students have the same birthday is more than 50%.
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.
A family went to the bridge at night. The dad can cross it in 1 minute, the mum in 2 minutes, the child in 5 minutes, and the grandmother in 10 minutes. They have one flashlight. The bridge only withstands two people. How can they cross the bridge in 17 minutes? (If two people cross, then they pass with the lower of the two speeds. They cannot pass along the bridge without a flashlight. They cannot shine the light from afar. They cannot carry anyone in their arms. They cannot throw the flashlight.)
Prove that the infinite decimal \(0.1234567891011121314 \dots\) (after the decimal point, all of the natural numbers are written out in order) is an irrational number.
There are \(n\) cities in a country. Between each two cities an air service is established by one of two airlines. Prove that out of these two airlines at least one is such that from any city you can get to any other city whilst traveling on flights only of this airline.