**A. 585.** For some integer *n*2 each pair (*i*,*j*), where 1*i*,*j**n*, is written on a card. We play the following game. The *n*^{2} cards are placed in an *n*×*n *table so that for every *i* and *j*, the card (*i*,*j*) is in the *i*th row, at the *j*th position. It is allowed to exchange the cards (*i*,*j*) and (*k*,*l*) if they are in the same row or in the same column, and *i*=*k* or *j*=*l*. Is it possible to reach that arrangement of the cards which contains (1,2) and (2,1) interchanged and all other cards are at their initial positions?

Based on the idea of *Zoltán Bertalan,* Békéscsaba

(5 points)

**A. 586.** Two convex quadrilaterals have side lengths *a*_{1}, *a*_{2}, *a*_{3}, *a*_{4}, and *A*_{1}, *A*_{2}, *A*_{3}, *A*_{4}; their areas are *t* and *T*, respectively. Show that

(The indices are used modulo 4.)

(5 points)

**B. 4527.** In a dice game, the player rolls three dice simultaneously, and then he may roll two more times any number of his dice (0, 1, 2 or 3). The player wins the game if all three dice have the same number on top after the last roll. What is the best strategy, and with that strategy what is the probability of winning?

Suggested by *E. Gáspár Merse,* Budapest

(5 points)

**C. 1162.** A circle is drawn over the shorter diagonal *AC*=*a* of a parallelogram *ABCD*. The intersections of the circle and the parallelogram form the hexagon *AIJCKL*, whose sides have lengths , *b*, *b*, , *b*, *b* in this order. Determine the sides and angles of the parallelogram.

(5 points)

**C. 1163.** For what number pairs *x*, *y* will the numbers *x*, *x*^{log x}, *y*^{log y}, be consecutive terms of a geometric progression? (log denotes decimal logarithm.)

Suggested by *A. Balga,* Budapest

(5 points)

**K. 377.** A 100×100 cm square tablecloth is laid on a 120×120 cm square table, such that the centres of the two squares coincide, the sides of the tablecloth are parallel to the diagonals of the tabletop, and the corners of the tablecloth are hanging over the edges of the table. What is the area of the uncovered part of the tabletop?

(6 points)

This problem is for grade 9 students only.

**K. 378.** Some friends had dinner in a restaurant, and decided to give a 15% tip to the waiter. If everyone pays 6000 forints (HUF, Hungarian currency) then the sum will be 2800 forints more than the price of the food but it will not cover the tip. If everyone pays 6600 forints then the sum will cover the tip, too, and they will get 1120 forints back. What was the total on the bill, and how many friends had dinner?

(6 points)

This problem is for grade 9 students only.