A. 534. The sides of a triangle are a, b and c, the lengths of the corresponding medians are sa, sb and sc, respectively. Prove that .
(Proposed by: Donát Nagy, Szeged)
A. 535. There is given a simple graph G with vertices v1,...,vn. H1,...,Hn are sets of nonnegative integers such that for every i=1,2,...,n, the cardinality of Hi is at most half of the degree of vi. Prove that G contains a subgraph G' with the same vertices such that the degree of vi in G' is not an element of Hi for any i.
(Proposed by: László Miklós Lovász, Budapest)