Subcontests
(3)an incantation contest
Two persons, A and B, set up an incantation contest in which they spell incantations (i.e. a finite sequence of letters) alternately. They must obey the following rules:
i) Any incantation can appear no more than once;
ii) Except for the first incantation, any incantation must be obtained by permuting the letters of the last one before it, or deleting one letter from the last incantation before it;
iii)The first person who cannot spell an incantation loses the contest. Answer the following questions:
a) If A says 'STAGEPREIMO' first, then who will win?
b) Let M be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are 2009 and containing only four letters A,B,C,D, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in M such that A has a winning strategy by starting with it. two triangles have equal circumradii
Two circles O1 and O2 intersect at M,N. The common tangent line nearer to M of the two circles touches O1,O2 at A,B respectively. Let C,D be the symmetric points of A,B with respect to M respectively. The circumcircle of triangle DCM intersects circles O1 and O2 at points E,F respectively which are distinct from M. Prove that the circumradii of the triangles MEF and NEF are equal.