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.
There are two purses and one coin. Inside the first purse is one coin, and inside the second purse is one coin. How can this be?
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).
A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.
Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.
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?
The following text is obtained by encoding the original message using Caesar Cipher.
WKHVLAWKROBPSLDGRIFUBSWRJUDSKBGHGLFDWHGWKHWRILIWLHWKBHDURIWKHEULWLVKVHFUHWVHUYLFH.
The following text is also obtained from the same original text:
KYVJZOKYFCPDGZRUFWTIPGKFXIRGYPUVUZTRKVUKYVKFWZWKZVKYPVRIFWKYVSIZKZJYJVTIVKJVIMZTV.
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 \(1/1000\).