MathDB

2009 Italy TST

Part of Italy TST

Subcontests

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All pairs of integers x, y satisfying equation

Find all pairs of integers (x,y)(x,y) such that y3=8x6+2x3y−y2. y^3=8x^6+2x^3y-y^2.

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 '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.
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n lamps on a circle

Let n,kn,k be positive integers such that n≥kn\ge k. nn lamps are placed on a circle, which are all off. In any step we can change the state of kk consecutive lamps. In the following three cases, how many states of lamps are there in all 2n2^n possible states that can be obtained from the initial state by a certain series of operations? i)kk is a prime number greater than 22; ii) kk is odd; iii) kk is even.

a polynomial with all real roots

Let nn be an even positive integer. An nn-degree monic polynomial P(x)P(x) has nn real roots (not necessarily distinct). Suppose yy is a positive real number such that for any real number t<yt<y, we have P(t)>0P(t)>0. Prove that P(0)1n−P(y)1n≥y.P(0)^{\frac{1}{n}}-P(y)^{\frac{1}{n}}\ge y.