Problems

The function \(f(x)\) on the interval \([a, b]\) is equal to the maximum of several functions of the form \(y = C \times 10^{- | x-d |}\) (where \(d\) and \(C\) are different, and all \(C\) are positive). It is given that \(f (a) = f (b)\). Prove that the sum of the lengths of the sections on which the function increases is equal to the sum of the lengths of the sections on which the function decreases.

An after school club was attended by 60 pupils. It turns out that in any group of 10 there will always be 3 classmates. Prove that within the group of 60 who attended there will always be at least 15 pupils from the same class.

Some points with integer co-ordinates are marked on a Cartesian plane. It is known that no four points lie on the same circle. Prove that there will be a circle of radius 1995 in the plane, which does not contain a single marked point.

Let \(n\) numbers are given together with their product \(p\). The difference between \(p\) and each of these numbers is an odd number.

Prove that all \(n\) numbers are irrational.

Are there such irrational numbers \(a\) and \(b\) so that \(a > 1\), \(b > 1\), and \(\lfloor a^m\rfloor\) is different from \(\lfloor b^n\rfloor\) for any natural numbers \(m\) and \(n\)?

This is a famous problem, called Monty Hall problem after a popular TV show in America.
In the problem, you are on a game show, being asked to choose between three doors. Behind each door, there is either a car or a goat. You choose a door. The host, Monty Hall, picks one of the other doors, which he knows has a goat behind it, and opens it, showing you the goat. (You know, by the rules of the game, that Monty will always reveal a goat.) Monty then asks whether you would like to switch your choice of door to the other remaining door. Assuming you prefer having a car more than having a goat, do you choose to switch or not to switch?

Can three points with integer coordinates be the vertices of an equilateral triangle?

Definition A set is a collection of elements, containing only one copy of each element. The elements are not ordered, nor they are governed by any rule. We consider an empty set as a set too.
There is a set \(C\) consisting of \(n\) elements. How many sets can be constructed using the elements of \(C\)?

A rectangular parallelepiped of the size \(m\times n\times k\) is divided into unit cubes. How many rectangular parallelepipeds are formed in total (including the original one)?

In the Land of Linguists live \(m\) people, who have opportunity to speak \(n\) languages. Each person knows exactly three languages, and the sets of known languages may be different for different people. It is known that \(k\) is the maximum number of people, any two of whom can talk without interpreters. It turned out that \(11n \leq k \leq m/2\). Prove that then there are at least \(mn\) pairs of people in the country who will not be able to talk without interpreters.