Problems

At the disposal of a tile layer there are 10 identical tiles, each of which consists of 4 squares and has the shape of the letter L (all tiles are oriented the same way). Can he make a rectangle with a size of \(5 \times 8\)? (The tiles can be rotated, but you cannot turn them over). For example, the figure shows the wrong solution: the shaded tile is incorrectly oriented.

A square napkin was folded in half, the resulting rectangle was then folded in half again (see the figure). The resulting square was then cut with scissors (in a straight line). Could the napkin have been broken up a) into 2 parts? b) into 3 parts? c) into 4 parts? d) into 5 parts? If yes – illustrate such a cut, if not – write the word “no”.

The number \(A\) is positive, \(B\) is negative, and \(C\) is zero. What is the sign of the number \(AB + AC + BC\)?

Find the area of the figures shown below.

Prove that if 21 people collected 200 nuts between them, there are two people in the group who collected the same number of nuts.

a) Prove that in any football team there are two players who were born on the same day of the week.

b) Prove that in the population of London, which is almost 9 million, there will be ten thousand people who celebrate their birthday on the same day.

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

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.

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.)

On Brennan’s birthday, the postman Daniel wants to find out how old Brennan is. Sarah says that Brennan is over 11 years old, and Matt claims that he is more than 10 years old. How old is Brennan, if it is known that exactly one of them was mistaken? Justify your answer.