Problems

In a row there are 2023 numbers. The first number is 1. It is known that each number, except the first and the last, is equal to the sum of two neighboring ones. Find the last number.

2022 dollars were placed into some wallets and the wallets were placed in some pockets. It is known that there are more wallets in total than there are dollars in any pocket. Is it true that there are more pockets than there are dollars in one of the wallets? You are not allowed to place wallets one inside the other.

Two players in turn paint the sides of an \(n\)-gon. The first one can paint the side that borders either zero or two colored sides, the second – the side that borders one painted side. The player who can not make a move loses. At what \(n\) can the second player win, no matter how the first player plays?

In a basket, there are 30 mushrooms. Among any 12 of them there is at least one brown one, and among any 20 mushrooms, there is at least one chanterelle. How many brown mushrooms and how many chanterelles are there in the basket?

100 cars are parked along the right hand side of a road. Among them there are 30 red, 20 yellow, and 20 pink Mercedes. It is known that no two Mercedes of different colours are parked next to one another. Prove that there must be three Mercedes cars parked next to one another of the same colour somewhere along the road.

20 birds fly into a photographer’s studio – 8 starlings, 7 wagtails and 5 woodpeckers. Each time the photographer presses the shutter to take a photograph, one of the birds flies away and doesn’t come back. How many photographs can the photographer take to be sure that at the end there will be no fewer than 4 birds of one species and no less than 3 of another species remaining in the studio.

Welcome back! We hope you all had a great summer and now you are ready for the new school year full of fun problems in mathematics. We decided to start with warm-up topic called dissections, so today we will cut various shapes into more elaborate geometric figures in order to reassemble them into a different shape.

Cut the following shape into two equal parts.

Cut the following shape into four equal figures.

Daniel has drawn on a sheet of paper a circle and a dot inside it. Show that he can cut a circle into two parts which can be used to make a circle in which the marked point would be the center.