MathDB
Very strange inequality

Source: 1996 IMO Shortlist

June 11, 2006
inequalitiesalgebrapolynomialfunctioncalculusIMO Shortlist

Problem Statement

Let a1,a2...an a_{1}, a_{2}...a_{n} be non-negative reals, not all zero. Show that that (a) The polynomial p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n} has preceisely 1 positive real root R R. (b) let A \equal{} \sum_{i \equal{} 1}^n a_{i} and B \equal{} \sum_{i \equal{} 1}^n ia_{i}. Show that AA≤RB A^{A} \leq R^{B}.
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