# 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.