Problems

On the diagram below \(AD\) is the bisector of the triangle \(ABC\). The point \(E\) lies on the side \(AB\), with \(AE = ED\). Prove that the lines \(AC\) and \(DE\) are parallel.

On the diagram below the line \(BD\) is the bisector of the angle \(\angle ABC\) in the triangle \(ABC\). A line through the vertex \(C\) parallel to the line \(BD\) intersects the continuation of the side \(AB\) at the point \(E\). Find the angles of the triangle \(BCE\) triangle if \(\angle ABC = 110^{\circ}\).

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 \(C\)?

Given a natural number \(n\) you are allowed to perform two operations: "double up", namely get \(2n\) from \(n\), and "increase by \(1\)", i.e. to get \(n+1\) from \(n\). Find the smallest amount of operations one needs to perform to get the number \(n\) from \(1\).

In a certain state, there are three types of citizens:

• A fool considers everyone a fool and themselves smart;

• A modest clever person knows truth about everyone’s intellectual abilities and consider themselves a fool;

• A confident clever person knows about everyone intellectual abilities correctly and consider themselves smart.

There are \(200\) deputies in the High Government. The Prime Minister conducted an anonymous survey of High Government members, asking how many smart people are there in the High Government. After reading everyone’s response he could not find out the number of smart people. But then the only member who did not participate in the survey returned from the trip. They filled out a questionnaire about the entire Government including themselves and after reading it the Prime Minister understood everything. How many smart could there be in the High Government (including the traveller)?

There are \(100\) people standing in line, and one of them is Arthur. Everyone in the line is either a knight, who always tells the truth, or a liar who always lies. Everyone except Arthur said, "There are exactly two liars between Arthur and me." How many liars are there in this line, if it is known that Arthur is a knight?

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.

Each integer on the number line is coloured either white or black. The numbers \(2016\) and \(2017\) are coloured differently. Prove that there are three identically coloured integers which sum to zero.

There are \(100\) non-zero numbers written in a circle. Between every two adjacent numbers, their product was written, and the previous numbers were erased. It turned out that the number of positive numbers after the operation coincides with the amount of positive numbers before. What is the minimum number of positive numbers that could have been written initially?