Zi Song's iterated polynomial

Algebra Level 5

For all ordered triples (p,q,r) (p,q,r), define the polynomial fp,q,r(x)=x3px2+qxr.f_{p, q, r}(x) = x^{3} - px^{2} + qx - r. Let a1,a2,a3,b1,b2,b3,c1,c2,c3a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3 be (not necessary distinct) positive real numbers such that the roots of fa1,a2,a3(x)f_{a_1, a_2, a_3}(x) are b1,b2,b3b_1, b_2, b_3 and the roots of fb1,b2,b3(x)f_{b_1, b_2, b_3}(x) are c1,c2,c3c_1, c_2, c_3 . What is the maximum possible value of 9b33b1+3+4+3b1+2b2+b3a1+1?\frac{9\sqrt[3]{b_3}}{b_1 + 3} + \frac{4+3b_1 + 2b_2 + b_3}{a_1 + 1}?

This problem is posed by Zi Song Y.

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