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polynomial to be killed

Source: Indonesia IMO 2010 TST, Stage 1, Test 5, Problem 2

November 12, 2009
algebrapolynomialinequalitiesalgebra proposed

Problem Statement

Consider a polynomial with coefficients of real numbers \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d with three positive real roots. Assume that ϕ(0)<0 \phi(0)<0, prove that 2b^3\plus{}9a^2d\minus{}7abc \le 0. Hery Susanto, Malang