Problems

Cut the board shown in the figure into four congruent parts so that each of them contains three shaded cells. Where the shaded cells are placed in each part need not be the same.

Gerard says: the day before yesterday I was 10 years old, and next year I will turn 13. Can this be?

We call a natural number “amazing” if it has the form \(a^b + b^a\) (where \(a\) and \(b\) are natural numbers). For example, the number 57 is amazing, since \(57 = 2^5 + 5^2\). Is the number 2006 amazing?

Homework. Cut a hole in an exercise book of a size so that you yourself can climb through it.

a) In how many ways can Dima paint five Christmas trees in silver, green and blue colours, if the amount of paint is unlimited, and he paints each tree in only one colour?

b) Dima has five baubles: a red, a green, a yellow, a blue and a gold one. In how many ways can he decorate five Christmas trees with them, if he needs to put exactly one bauble on each tree?

c) What about if he can hang several baubles on one Christmas tree (and all of the baubles have to be used)?

Between them, Jennifer and Alex shared the money they made from running a lemonade stand. Jennifer thought: “If I took \(40\%\) more money then Alex’s share would decrease by \(60\%\)”. How would Alex’s share of the profits change if Jennifer took \(50\%\) more money for herself?

Henry did not manage to get into the elevator on the first floor of the building and decided to go up the stairs. It takes 2 minutes to rise to the third floor. How long does it take to rise to the ninth floor?

In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, \(\frac{1}{2}\) a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.

A teacher filled the squares of a chequered table with \(5\times5\) different integers and gave one copy of it to Janine and one to Zahara. Janine selects the largest number in the table, then she deletes the row and column containing this number, and then she selects the largest number of the remaining integers, then she deletes the row and column containing this number, etc. Zahara performs similar operations, each time choosing the smallest numbers. Can the teacher fill up the table in such a way that the sum of the five numbers chosen by Zahara is greater than the sum of the five numbers chosen by Janine?

• Eight schoolchildren solved \(8\) tasks. It turned out that \(5\) schoolchildren solved each problem. Prove that there are two schoolchildren, who solved every problem at least once.

• If each problem is solved by \(4\) pupils, prove that it is not necessary to have two schoolchildren who would solve each problem.