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Part of 2011 Mediterranean Mathematics Olympiad

Problems(1)

Mediterranean Polynomials

Source: MMO 2011 Problem #1

9/11/2011
A Mediterranean polynomial has only real roots and it is of the form P(x)=x10−20x9+135x8+a7x7+a6x6+a5x5+a4x4+a3x3+a2x2+a1x+a0 P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 with real coefficients a0…,a7a_0\ldots,a_7. Determine the largest real number that occurs as a root of some Mediterranean polynomial. (Proposed by Gerhard Woeginger, Austria)
algebrapolynomialVietainequalitiesalgebra unsolved