**B. 4442.** Ann, Bill and Cecilia play the following game. They take turns choosing an integer between 1 and 10, and adding that number to the sum of the numbers that have been said so far. (For simplicity, in each turn, the player states the sum.) The player who is first able to say 100 wins the game. Prove that the two girls, with the appropriate strategy, are able to achieve that one of them is the winner.

Suggested by *B. Futó,* New York

(3 points)

**B. 4443.** *a*_{1},*a*_{2},...,*a*_{n} és *b*_{1},*b*_{2},...,*b*_{k} are two given sequences with positive integer terms with *a*_{i}*k* and *b*_{j}*n*. Show that there are integers 0*i*_{1}<*i*_{2}*n* and 0*j*_{1}<*j*_{2}*k* such that .

(5 points)

**B. 4444.** The points *E* and *G*, respectively, are obtained by rotating vertex *D* of a square *ABCD* about *A* and vertex *B* (lying opposite *D*) about *C* outwards, through the same acute angle . The intersection of lines *AE* and *BG *is *F*. Line *GD* intersects the circles *GCB* and *BDE* again at points *L *and *M*, respectively. Prove that the points *A*, *B*, *F*, *L*, *M* are cyclic.

Suggested by *J. Bodnár*

(3 points)