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.
In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.
Every point in a plane, which has whole-number co-ordinates, is plotted in one of \(n\) colours. Prove that there will be a rectangle made out of 4 points of the same colour.
One of \(n\) prizes is embedded in each chewing gum pack, where each prize has probability \(1/n\) of being found.
How many packets of gum, on average, should I buy to collect the full collection prizes?
On a \(100 \times 100\) board 100 rooks are placed that cannot capturing one another.
Prove that an equal number of rooks is placed in the upper right and lower left cells of \(50 \times 50\) squares.