**A. 404.** The vertices of a regular 2*n*-gon are . Call a diagonal *V*_{i}*V*_{j} *even* if *i* and *j* have the same parity. Dissect the polygon into triangles arbitrarily drawing 2*n*-3 nonintersecting diagonals. The following operation is allowed on this dissection: Choose two vertices, *V*_{i} and *V*_{j}, which are either consecutive or they are connected by a diagonal used for the dissection. Then, on one side of the line *V*_{i}*V*_{j} replace each diagonal by its mirror image through the perpendicular bisector of *V*_{i}*V*_{j} (see the *figure*). Prove that, starting from an arbitrary dissection and applying this operation several times, it can be achieved that all even diagonals used for the dissection connect only vertices of even indices.

(Based on the sixth problem of the *47th IMO,* Slovenia)

(5 points)

**A. 405.** The real numbers *a*, *b*, *c*, *x*, *y*, *z* satisfy *a**b**c*>0 and *x**y**z*>0. Prove that

(*Korean competition problem*)

(5 points)

**B. 3926.** Ann and Bob are playing the following game: There are 10 heaps of pebbles in front of them. The first heap contains 1 pebble, the second one contains 2, the third one contains 3, and so on, the tenth heap contains 10 pebbles. They take turns in carrying out one of the following two moves: they either divide a heap into two smaller ones or remove one pebble from one heap. A player who cannot do so any more will lose the game. Determine which player has a winning strategy, given that Ann starts the game.

(5 points)

**B. 3927.** Vertex *A* of a tetrahedron *ABCD* is reflected about *B*, *B* is reflected about *C*, *C *about *D* and *D* about *A*. Let the respective reflections be *A*', *B*', *C*' and *D*'. By what factor is the volume of the tetrahedron *A*'*B*'*C*'*D*' larger than that of the original tetrahedron *ABCD*?

(Suggested by *G. Holló,* Budapest)

(4 points)

**K. 85.** Annie and Bennie were playing with a chair. When Annie was standing on the chair and Bennie on the floor, the top of Annie's head was 30 cm higher than Bennie's. When they exchanged places, Bennie standing on the chair and Annie on the floor, the top of Bennie's head was half a metre higher than Annie's. How tall is the chair?

(6 points)

This problem is for grade 9 students only.

**K. 87.** The growth of an isosceles Pythagoras tree has the following pattern: In the first year, the tree grows its trunk, which is a square. In the second year, an isosceles right-angled triangle grows on top, such that its hypotenuse is the top side of the square, and then the first two branches, also square in shape, grow from the legs of the triangle. Then this pattern repeats every year, that is, an isosceles right triangle grows on the top of each branch and its legs grow two new square branches. Given that the trunk (i.e. the first square) is 8 metres wide, calculate the height and the width of the tree at the end of the fourth year. (Hint: Draw a diagram of the tree to scale on squared paper. Let one little square represent 2 metres.)

(6 points)

This problem is for grade 9 students only.

**K. 88.** There are four ninth-grade classes at a school. The four classes went on a four-day camp. Each class went hiking on one of the four days, while the other classes took part in presentations and other activities. On Monday, class *A* went hiking and thus 81 students remained in the camp. On Tuesday, class *B* were hiking and thus 79 students remained in the camp. On Wednesday, class *C* were hiking and 75 students stayed in the camp. Finally, on Thursday, with class *D* hiking, 80 students stayed in the camp. How many students are there in each class?

(6 points)

This problem is for grade 9 students only.