# Zi Song's iterated polynomial

Algebra Level 5

For all ordered triples $$(p,q,r)$$, define the polynomial $f_{p, q, r}(x) = x^{3} - px^{2} + qx - r.$ Let $$a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$$ be (not necessary distinct) positive real numbers such that the roots of $$f_{a_1, a_2, a_3}(x)$$ are $$b_1, b_2, b_3$$ and the roots of $$f_{b_1, b_2, b_3}(x)$$ are $$c_1, c_2, c_3$$. What is the maximum possible value of $\frac{9\sqrt[3]{b_3}}{b_1 + 3} + \frac{4+3b_1 + 2b_2 + b_3}{a_1 + 1}?$

This problem is posed by Zi Song Y.

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