HMMT 2013 #0jeq

HMMT 2013Hard

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Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Vieta's formulasRoots of unityComplex numbersSymmetric functionsPolynomial operations

Problem

Problem:

The polynomial $f(x) = x^{3} - 3x^{2} - 4x + 4$ has three real roots $r_{1}$ , $r_{2}$ , and $r_{3}$ . Let $g(x) = x^{3} + a x^{2} + b x + c$ be the polynomial which has roots $s_{1}$ , $s_{2}$ , and $s_{3}$ , where

$s_{1} = r_{1} + r_{2} z + r_{3} z^{2}$ ,

$s_{2} = r_{1} z + r_{2} z^{2} + r_{3}$ ,

$s_{3} = r_{1} z^{2} + r_{2} + r_{3} z$ ,

and $z = \frac{-1 + i \sqrt{3}}{2}$ .

Find the real part of the sum of the coefficients of $g(x)$ .

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