MathDB

3

Part of 2009 Italy TST

Problems(2)

All pairs of integers x, y satisfying equation

Source: Italy TST 2009 P3

3/10/2012
Find all pairs of integers (x,y)(x,y) such that y3=8x6+2x3yāˆ’y2. y^3=8x^6+2x^3y-y^2.
geometrynumber theory proposednumber theory
an incantation contest

Source: Italy TST 2009 p6

3/10/2012
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 'STAGEPREIMOSTAGEPREIMO' first, then who will win? b) Let MM be the set of all possible incantations whose lengths (i.e. the numbers of letters in them) are 20092009 and containing only four letters A,B,C,DA,B,C,D, each of them appearing at least once. Find the first incantation (arranged in dictionary order) in MM such that A has a winning strategy by starting with it.
modular arithmeticcombinatorics proposedcombinatorics