# Samir's Cubic Roots

Algebra Level 5

Suppose $$a$$ and $$b$$ are positive integers satisfying $$1 \leq a \leq 31$$, $$1 \leq b \leq 31$$ such that the polynomial $$P(x)=x^3-ax^2+a^2b^3x+9a^2b^2$$ has roots $$r$$, $$s$$, and $$t$$.

Given that there exists a positive integer $$k$$ such that $$(r+s)(s+t)(r+t)=k^2$$, compute the maximum possible value of $$ab$$.

This problem is posed by Samir K.

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