Problem 33

Algebra Level 5

\[\large \dfrac{5a^2-3ab+2}{a^2b-a^3}\]

Let \(a\) and \(b\) be non-zero real numbers such that all the roots of the equation \(ax^3-x^2+bx-1=0\) are positive reals.

Given that the minimum value of the expression above can be expressed as \(H\sqrt{K}\) and the equality achieved when \[a=\dfrac{A\sqrt{B}}{C}, \qquad b=\sqrt{S}\] where \(H,K,A,B, C\) and \(S\) are positive integers with \(K,B,S\) being square-free numbers and \(\gcd(A,C)=1\).

Compute \(H+K+A+B+C+S\).


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