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inequality on coefficients of cubic

Source: (Indian) RMO 2008 Problem 3

November 9, 2008

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Problem Statement

Suppose

a

and

b

are real numbers such that the roots of the cubic equation ax^3\minus{}x^2\plus{}bx\minus{}1 are positive real numbers. Prove that:

(i)\ 0<3ab\le 1\text{ and }(i)\ b\ge \sqrt{3}

[19 points out of 100 for the 6 problems]