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.

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.

A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.