**B. 3832.** *P* is an arbitrary point of the hypotenuse *AB* of a right-angled triangle *ABC*. The foot of the altitude drawn from vertex *C* is *C*_{1}. The projection of *P* onto the leg *AC* is *A*_{1}, and its projection onto the leg *BC* is *B*_{1}.

*a*) Prove that the points *P*, *A*_{1}, *C*, *B*_{1}, *C*_{1} lie on a circle.

*b*) Prove that the triangles *A*_{1}*B*_{1}*C*_{1} and *ABC* are similar.

(3 points)

**B. 3841.** The problem below appears in the article ``Szalonpóker'' (Hungarian title) of the current issue. Suppose that a deck of 52 cards is suffled in the same way. How many times does it need to be shuffled in order to get back the initial order. Solve the problem for that case, too, when the shuffling starts with the bottom card of the deck on the right, that is, when the card in the 26th place originally is put at the bottom of the new deck.

(Little John and Old Firehand drop in the famous casino of Black Jacky to play cards. They play with a deck of 32 cards numbered 1 to 32. Before they agree on the rules, Black Jacky shuffles the cards as follows: He places the deck on the table, removes the top 16 cards and puts them on the table, to the right of the remaining deck without turning over. Then he forms one deck of them, with the bottom card of the deck on the left lying at the bottom, followed by alternating cards from the two decks. Then he repeats the procedure several times with the deck obtained in this way. Little John is sure that this way of shuffling the cards is not fair. Show that repeating the procedure several times will lead to a surprising result.)

(4 points)

**K. 43.** Two students are talking after class. ``What is the average of your computer science marks in September?

-- 4.6 exactly.

-- That is impossible. The school year has just started, you cannot have so many marks yet.''

What may the student in doubt have thought of? (The grades in Hungary are varying from 1 to 5.) (Based on the idea of

(6 points)

This problem is for grade 9 students only.

**K. 46.** The values of a gold artefact in Aurum is proportional to the square of its mass. A 100-dollar piece is stolen by thieves. The thieves cut it up into smaller pieces of equal mass to prepare pendants of them, whose total value is 10 dollars. The pendants are bought up by a jeweller who assembles them into bracelets (of not necessarily the same mass). How much are the individual bracelets worth, given that each bracelet is made of a whole number of pendants, and their total value is 46 dollars?

(6 points)

This problem is for grade 9 students only.

**K. 47.** There are two kinds of people living on an island: the good and the bad. The good always tell the truth and the bad always lie. Naturally, every inhabitant of the island is either a boy or a girl. Here is a conversation of two young people on the island:

*A*: ``If I am good, *B* is bad.''

*B*: ``If I am a boy, *A* is a girl.''

Find out if each of them is good or bad and what sex they belong to.

(6 points)

This problem is for grade 9 students only.