IMO Shortlist 1996 problem A4


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2. travnja 2012.
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Let a_{1}, a_{2}...a_{n} be non-negative reals, not all zero. Show that that
(a) The polynomial p(x) = x^{n} - a_{1}x^{n - 1} + ... - a_{n - 1}x - a_{n} has preceisely 1 positive real root R.
(b) let A = \sum_{i = 1}^n a_{i} and B = \sum_{i = 1}^n ia_{i}. Show that A^{A} \leq R^{B}.
Izvor: Međunarodna matematička olimpijada, shortlist 1996