**B. 4552.** In this sentence, the number of numbers occurring once is *a*_{1}, the number of numbers occurring twice is *a*_{2}, ..., and the number of numbers occurring 2013 times is *a*_{2013}. Determine the values of such that the resulting statement is true. In how many different ways can that be done?

Suggested by *L. Kozma,* Saarbrücken, Germany

(5 points)

**B. 4559.** The interior angle bisectors drawn from vertices *A*, *B* and *C* of triangle *ABC *intersect the circumscribed circle at the points *D*, *E* and *F*, respectively. The intersections of the sides of triangles *DEF* and *ABC*, starting from *A*, towards *B*, are *G*, *H*, *I*, *J*, *K* and *L* in this order. Show that the triangles *DGL*, *EHI* and *FKJ* are all similar.

Suggested by *Sz. Miklós,* Herceghalom

(6 points)

**B. 4560.** The network of roads in the city of Icosapolis corresponds to the edge graph of an icosahedron. The home of Iorgos is at one vertex of the icosahedron, while his favourite theater is situated at the opposite vertex. On his way home from the theatre after dark, he stops at every vertex he reaches, and after some hesitation decides which way to proceed. Assume that at any vertex the probability of meeting someone who will show him one possible direction to get him home along the least possible number of edges is *p*. Otherwise he will chose an edge at random. (He may as well turn back in the direction he came from.) For what value of *p* is there a 50% chance that he will get home before getting back to the theatre?

Suggested by *M. E. Gáspár,* Budapest

(6 points)

**C. 1176.** *a*, *b*, *c*, *d*, *e* are five consecutive integers in increasing order. The dimensions of a cuboid are *a*, *b*, *c*. For what values will the diagonal of the cuboid be the hypotenuse of a right-angled triangle with legs *d* and *e*?

(5 points)

This problem is for grade 1 - 10 students only.

**K. 379.** Kate sewed a button on her coat. The button has four holes in it as shown in the *figure* (the holes form the four vertices of a square). As the thread is pulled through the holes again and again, it produces various patterns, as viewed from the front. One such pattern is shown in the *figure.* How many different patterns may result, provided that at least two holes need to be used to fix the button to the coat?

(6 points)

This problem is for grade 9 students only.

**K. 382.** Use the digits 9, 8, 8, 7, 7, 7 and one more digit of your choice to write down the largest seven-digit number divisible by 36.

(6 points)

This problem is for grade 9 students only.