Problems

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.

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.

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?