IMO Shortlist 2010 C7

Stefan Spataru
Mesaje: 84
Membru din: Dum Mar 27, 2011 10:36 pm

IMO Shortlist 2010 C7

Mesaj de Stefan Spataru »

Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied:

(i) each integer belongs to at least one of them;
(ii) each progression contains a number which does not belong to other progressions.

Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization.

Prove that $\[s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).\]$
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