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).
2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.
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.
The key of the cipher, called the “swivelling grid”, is a stencil made from a square sheet of chequered paper of size \(n \times n\) (where \(n\) is even). Some of the cells are cut out. One side of the stencil is marked. When this stencil is placed onto a blank sheet of paper in four possible ways (marked side up, right, down or left), its cut-outs completely cover the entire area of the square, where each cell is found under the cut-out exactly once. The letters of the message, that have length \(n^2\), are successively written into the cut-outs of the stencil, where the sheet of paper is placed on a blank sheet of paper with the marked side up. After filling in all of the cut-outs of the stencil with the letters of the message, the stencil is placed in the next position, etc. After removing the stencil from the sheet of paper, there is an encrypted message.
Find the number of different keys for an arbitrary even number \(n\).
Let \(x\) be a natural number. Among the statements:
\(2x\) is more than 70;
\(x\) is less than 100;
\(3x\) is greater than 25;
\(x\) is not less than 10;
\(x\) is greater than 5;
three are true and two are false. What is \(x\)?
A city in the shape of a triangle is divided into 16 triangular blocks, at the intersection of any two streets is a square (there are 15 squares in the city). A tourist began to walk around the city from a certain square and travelled along some route to some other square, whilst visiting every square exactly once. Prove that in the process of travelling the tourist at least 4 times turned by \(120^{\circ}\).
26 numbers are chosen from the numbers 1, 2, 3, ..., 49, 50. Will there always be two numbers chosen whose difference is 1?
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.
Prove that there is no polyhedron that has exactly seven edges.