Your task is to find out a five-digit phone number, asking questions that can be answered with either “yes” or “no.” What is the smallest number of questions for which this can be guaranteed (provided that the questions are answered correctly)?
Upon the installation of a keypad lock, each of the 26 letters located on the lock’s keypad is assigned an arbitrary natural number known only to the owner of the lock. Different letters do not necessarily have different numbers assigned to them. After a combination of different letters, where each letter is typed once at most, is entered into the lock a summation is carried out of the corresponding numbers to the letters typed in. The lock opens only if the result of the summation is divisible by 26. Prove that for any set of numbers assigned to the 26 letters, there exists a combination that will open the lock.
In an \(n\) by \(n\) grid, \(2n\) of the squares are marked. Prove that there will always be a parallelogram whose vertices are the centres of four of the squares somewhere in the grid.
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 2001.
A hostess bakes a cake for some guests. Either 10 or 11 people can come to her house. What is the smallest number of pieces she needs to cut the cake into (in advance) so that it can be divided equally between 10 and 11 guests?
How many ways can I schedule the first round of the Russian Football Championship, in which 16 teams are playing? (It is important to note who is the host team).
Two players play the following game. They take turns. One names two numbers that are at the ends of a line segment. The next then names two other numbers, which are at the ends of a segment nested in the previous one. The game goes on indefinitely. The first aims to have at least one rational number within the intersection of all of these segments, and the second aims to prevent such occurring. Who wins in this game?
2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.
Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).
Father Christmas has an infinite number of sweets. A minute before the New Year, Father Christmas gives some children 100 sweets, while the Snow Maiden takes one sweet from them. Within half a minute before the New Year, Father Christmas gives the children 100 more sweets, and the Snow Maiden again takes one sweet. The same is repeated for 15 seconds, for 7.5 seconds, etc. until the new Year. Prove that the Snow Maiden will be able to take away all the sweets from the children by the New Year.