Problems

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)?

A toddler has \(25\) lego pieces in a box:

• In how many ways are there to choose three pieces to play with?

• In how many ways can he choose three pieces for the foundation, main walls and roof? Note that the order is important.

Cut the shape (see the figure) into two identical pieces (coinciding when placed on top of one another).

Gabby is standing on a river bank. She has two clay jars: one – for 5 litres, and about the second Gabby remembers only that it holds either 3 or 4 litres. Help Gabby determine the capacity of the second jar. (Looking into the jar, you cannot figure out how much water is in it.)

In the garden of Sandra and Lewis 2006 rose bushes were growing. Lewis watered half of all the bushes, and Sandra watered half of all the bushes. At the same time, it turned out that exactly three bushes, the most beautiful, were watered by both Sandra and Lewis. How many rose bushes have not been watered?

In a burrow there is a family of 24 mice. Every night exactly four of them are sent to the warehouse for cheese.

Could it occur that at some point in time each mouse went to the warehouse with every other mouse exactly one time?

In a physics club, the teacher created the following experiment. He spread out 16 weights of weight 1, 2, 3, ..., 16 grams onto weighing scales, so that one of the bowls outweighed the other. Fifteen students in turn left the classroom and took with them one weight each, and after each student’s departure, the scales changed their position and outweighed the opposite bowl of the scales. What weight could remain on the scales?

Solve the equation: \(|x-2005| + |2005-x|=2006\).

There is a \(5\times 9\) rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?

Solve the equation: \[x + \frac{x}{x} + \frac{x}{x+\frac{x}{x}} = 1\]