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.

There is a rectangular table. Two players start in turn to place on it one pound coin each, so that these coins do not overlap one another. The player who cannot make a move loses. Who will win with the correct strategy?

a) Two players play in the following game: on the table there are 7 two pound coins and 7 one pound coins. In a turn it is allowed to take coins worth no more than three pounds. The one who takes the last coin wins. Who will win with the correct strategy?

b) The same question, if there are 12 one pound and 12 two pound coins.

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.

On Easter Island, people ask each other questions, to which only “yes” or “no” can be answered. In this case, each of them belongs exactly to one of the tribes either A or B. People from tribe A ask only those questions to which the correct answer is “yes”, and from tribe B – those questions to which the correct answer is “no.” In one house lived a couple Ethan and Violet Russell. When Inspector Krugg approached the house, the owner met him on the doorstep with the words: “Tell me, do Violet and I belong to tribe B?”. The inspector thought and gave the right answer. What was the right answer?

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