**A. 382.** *S* and *T* are disjoint sets, * is a binary operation on the elements of *S* and o is a binary operation on the elements of *T*. (That is, if *a*,*b**S* and *c*,*d**T*, then *a***b**S* and *c*o*d**T*). Each operation is associative. In other words, (*S*,*) and (*T*,o) are semigroups. It is also given that for every *t**T* there are elements *u*,*v**T*, such that *u*o*t*=*t*o*v*=*t*. Let denote an arbitrary mapping. Define the operation on the set *S**T* as follows:

Show that the operation is associative if and only if *f* is a homomorphism, that is, *f*(*a***b*)=*f*(*a*)o*f*(*b*) for all *a*,*b**S*.

Czech competition problem

(5 points)

**B. 3842.** Given are five 15-litre vessels, containing 1, 2, 3, 4, and 5 litres of water, respectively. We are allowed to double the amount of water in any chosen vessel from another one. What is the largest possible amount of water that can be collected in a vessel by repeating this step?

(3 points)

**K. 49.** 15 teams participated in a football tournament. Every team played each of the other teams once. 3 points were awarded for winning, 2 for a draw and 1 point for losing the game. At the end of the tournament, every team had a different number of points, 21 being the lowest score. Prove that the team with the highest score has played at least one draw.

Suggested by *B. Szalkai,* Veszprém

(6 points)

This problem is for grade 9 students only.