Problems

There were seven boxes. In some of them, seven more boxes were placed inside (not nested in each other), etc. As a result, there are 10 non-empty boxes. How many boxes are there now in total?

Four aliens – Dopey, Sleepy, Happy, Moody from the planet of liars and truth tellers had a conversation: Dopey to Sleepy: “you are a liar”; Happy to Sleepy: “you are a liar”; Moody to Happy: “Yes, they are both liars,” (after a moment’s thought), “however, so are you.” Which of them is telling the truth?

A class contains 25 pupils. It is known that within any group of 3 pupils there are two friends. Prove that there is a pupil who has no fewer than 12 friends.

A square area of size \(100\times 100\) is covered in tiles of size \(1\times 1\) in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?

One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?

It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.

In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?

One of five brothers baked a cake for their Mum. Alex said: “This was Vernon or Tom.” Vernon said: “It was not I and not Will who did it.” Tom said: “You’re both lying.” David said: “No, one of them told the truth, and the other was lying.” Will said: “No David, you’re wrong.” Mum knows that three of her sons always tell the truth. Who made the cake?

Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.