Problems

Colour the plane in red, green and blue in such a way that every line consists of points of at most two colours. Remember that you have to use all three colours.

I don’t know how the figure below can be made of several $1\times 5$ rectangles which do not overlap. I am willing to pay $1$ pound if you show me a possible way of doing that which I have not seen before. What is the maximal amount of money a person can earn by solving this problem?

a) What is the answer in case we are asked to split the figure below into $1\times 4$ rectangles instead of $1\times 5$ rectangles?

(b) In the context of Example 1 what is the answer in case we are asked to split the figure into $1\times 7$ rectangles instead of $1\times 5$ rectangles?

A $7\times 7$ square was tiled using $1\times 3$ rectangular blocks. One of the squares has not been covered. Which one can it be?

Ten pairwise distinct non-zero numbers are such that for each two of them either the sum of these numbers or their product is a rational number.

Prove that the squares of all numbers are rational.

On every cell of a $9\times 9$ board there is a beetle. At the sound of a whistle, every beetle crawls onto one of the diagonally neighbouring cells. Note that, in some cells, there may be more than one beetle, and some cells will be unoccupied.

Prove that there will be at least 9 unoccupied cells.

A grasshopper can make jumps of 8, 9 and 10 cells in any direction on a strip of $n$ cells. We will call the natural number $n$ jumpable if the grasshopper can, starting from some cell, bypass the entire strip, having visited each cell exactly once. Find at least one $n>50$ that is not jumpable.