**A. 534.** The sides of a triangle are *a*, *b* and *c*, the lengths of the corresponding medians are *s*_{a}, *s*_{b} and *s*_{c}, respectively. Prove that .

(Proposed by: *Donát Nagy,* Szeged)

(5 points)

**A. 535.** There is given a simple graph *G* with vertices *v*_{1},...,*v*_{n}. *H*_{1},...,*H*_{n} are sets of nonnegative integers such that for every *i*=1,2,...,*n*, the cardinality of *H*_{i} is at most half of the degree of *v*_{i}. Prove that *G* contains a subgraph *G*' with the same vertices such that the degree of *v*_{i} in *G*' is not an element of *H*_{i} for any *i*.

(Proposed by: *László Miklós Lovász,* Budapest)

(5 points)

**B. 4354.** Given four distinct points *A*, *B*, *C*, *D* on the plane, construct two circles touching each other on the outside such that one of them passes through *A* and *B*, the other passes through *C* and *D*, and their point of tangency lies on the line segment *BC*.

(4 points)