Problems

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 $11n\le k\le m/2$. 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?

Detective Nero Wolf investigates a crime. He’s got $80$ people involved in the case, among whom one is a criminal and another is a witness to the crime (but it is not known who either of them are). Each day the detective may invite one or more of these $80$ people, and if there is a witness among those invited, but not the perpetrator, the witness will report who the perpetrator is. Can the detective solve a case in $12$ days?

The king decided to reward a group of $n$ wise men. They will be placed in a row one after the other (so that everyone is looking in the same direction), and each is going to wear a black or a white hat. Everyone will see the hats of everyone in front, but not those behind them. The wise men will take turns (from the last to the first) to name the color (white or black) and the natural number of their choice.
At the end, the number of sages who have named the color of their hat correctly is counted: that is exactly how many days the whole group will be paid a salary raise. The wise men were allowed to agree in advance on how to respond. At the same time, the wise men know that exactly $k$ of them are insane (they do not know who exactly). Any insane man names the color white or black, regardless of the agreement. What is the maximum number of days with a pay supplement that the wise men can guarantee to a group, regardless of the location of the insane in the queue?

A whole number of litres of water were poured into three vessels. You can only to pour into any vessel the exact amount of water equal to the amount it already contains from any other vessel. Prove that in a few transfusions one can empty one of the vessels. The vessels are large enough: each can hold all the water.

Each integer on the number line is coloured either yellow or blue. Prove that there is a colour with the following property: For every natural number $k$, there are infinitely many numbers of this colour divisible by $k$.

A set includes weights weighing $1$ gram, $2$ grams, $4$ grams, ... (all powers of the number $2$), and in the set some of the weights might be the same. Weights were placed on two cups of the scales so that the scales are in balance. It is known that on the left cup, all weights are different. Prove that there are as many weights on the right cup as there are on the left.

There are $16$ cities in the kingdom. Prove that it is possible to build a system of roads in such a way that one can get from any city to any other without passing through more than one city on the way, and with at most five roads coming out of each city.

A graph is a finite set of points, some of which are connected with line segments. The points of a graph are called vertices. The line segments are called edges. In this problem set we only consider graphs in which every pair of vertices is connected with one or zero edges.

In a mathematical problem, one may use vertices of a graph to represent objects in the problem, i.e. people, cities, airports, and edges of the graph represent relations between the objects such as mutual friendship, railways between cities, plane routes. As you will see in the examples below, representing the initial problem as a graph can considerably simplify the solution.