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P(a)+P(b)+P(c)<= -1 if P(x)=3^{2005}x^{2007}- 3^{2005}x^{2006}-x^2,

Source: 2007 Indonesia TST stage 2 test 2 p3

December 14, 2020
algebrainequalitiespolynomial

Problem Statement

Let a,b,ca, b, c be positive reals such that a+b+c=1a + b + c = 1 and P(x)=32005x2007−32005x2006−x2P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2. Prove that P(a)+P(b)+P(c)≤−1P(a) + P(b) + P(c) \le -1.
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