Two people toss a coin: one tosses it 10 times, the other – 11 times. What is the probability that the second person’s coin showed heads more times than the first?
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.
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.