it is motivated by http://mathoverflow.net/questions/46249/density-of-congruence-classes-covered-by-a-set/46277#46277

is it true that for given positive integers $1 < b_1 < b_2 < \dots < b_k$ the union of regular $b_i$-gons inscribed to the same circle has the minimal cardinality when they share a vertex?
(here polygon is just the set of its vertices; 2-gon is diameter)