IMO Shortlist 1996 problem A2


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2. travnja 2012.
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Let a_1 \geq a_2 \geq \ldots \geq a_n be real numbers such that for all integers k > 0,

a^k_1 + a^k_2 + \ldots + a^k_n \geq 0.

Let p = max\{|a_1|, \ldots, |a_n|\}. Prove that p = a_1 and that

(x - a_1) \cdot (x - a_2) \cdots (x - a_n) \leq x^n - a^n_1 for all x > a_1.
Izvor: Međunarodna matematička olimpijada, shortlist 1996