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min P = (5a^{2}-3ab+2)/a^{2}(b-a)

Source: Mathcenter Contest / Oly - Thai Forum 2009 R2 p5 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 9, 2022

inequalitiesalgebra

Problem Statement

Let

a

and

b

be real numbers, where

a \not= 0

and

a \not= b

and all the roots of the equation

ax^{3}-x^{2}+bx-1 = 0

is a real and positive number. Find the smallest possible value of

P = \dfrac{5a^{2}-3ab+2}{a^{2}(b-a)}

.
(Heir of Ramanujan)