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the least real c

Source: 2012 China TST,Test 3,Problem 3

March 25, 2012
algebrapolynomialinequalitiesgeometrygeometric transformationalgebra unsolved

Problem Statement

Find the smallest possible value of a real number cc such that for any 20122012-degree monic polynomial P(x)=x2012+a2011x2011+…+a1x+a0P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0 with real coefficients, we can obtain a new polynomial Q(x)Q(x) by multiplying some of its coefficients by −1-1 such that every root zz of Q(x)Q(x) satisfies the inequality ∣Im⁡z∣≤c∣Re⁡z∣. \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert.
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