Problems

A equilateral triangle made of paper bends in a straight line so that one of the vertices falls on the opposite side as shown on the picture. Prove that the corresponding angles of the two white triangles are equal.

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)?

Winnie the Pooh has five friends, each of whom has pots of honey in their house: Tigger has \(1\) pot, Piglet has \(2\), Owl has \(3\), Eeyore has \(4\), and Rabbit has \(5\). Winnie the Pooh comes to visit each friend in turn, eats one pot of honey and takes the other pots with him. He came into the last house carrying \(10\) pots of honey. Whose house could Pooh have visited last?

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?

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 \(k\).