Three friends decide, by a coin toss, who goes to get the juice. They have one coin. How do they arrange coin tosses so that all of them have equal chances to not have to go and get the juice?
There are 8 glasses of water on the table. You are allowed to take any two of the glasses and make them have equal volumes of water (by pouring some water from one glass into the other). Prove that, by using such operations, you can eventually get all the glasses to contain equal volumes of water.
26 numbers are chosen from the numbers 1, 2, 3, ..., 49, 50. Will there always be two numbers chosen whose difference is 1?
A broken calculator carries out only one operation “asterisk”: \(a*b = 1 - a/b\). Prove that using this calculator it is possible to carry out all four arithmetic operations (addition, subtraction, multiplication, division).
There are 20 students in a class, and each one is friends with at least 14 others. Can you prove that there are four students in this class who are all friends?
A convex polygon on a plane contains no fewer than \(m^2+1\) points with whole number co-ordinates. Prove that within the polygon there are \(m+1\) points with whole number co-ordinates that lie on a single straight line.
All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.
A message is encrypted by replacing the letters of the source text with pairs of digits according to some table (known only to the sender and receiver) in which different letters of the alphabet correspond to different pairs of digits. The cryptographer was given the task to restore the encrypted text. In which case will it be easier for him to perform the task: if it is known that the first word of the second line is a “thermometer” or that the first word of the third line is “smother”? Justify your answer. (It is assumed that the cryptographic table is not known).
Prove that there is no polyhedron that has exactly seven edges.
Does a continuous function that takes every real value exactly 3 times exist?