Problems

What weights can three weights have so that they can weigh any integer number of kilograms from 1 to 10 on weighing scales (weights can be put on both cups)? Give an example.

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 $n$.

The rook stands on the square a1 of a chessboard. For a move, you can move it by any number of cells to the right or up. The one who puts the rook on the h8 square will win. Who wins with the right strategy?

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 ${120}^{\circ }$.

It is known that any person has at most 400,000 hairs on their head. Given that the population of London is not less than 8 million, prove that there are 20 Londoners with the same number of hairs on their heads.

Hannah has 10 employees. Each month, Hannah raises the salary by 1 pound of exactly nine of her employees (of her choice). How can Hannah raise the salaries to make them equal? (Salaries are an integer number of pounds.)

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?

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.