Samir's Cubic Roots

Algebra Level 5

Suppose aa and bb are positive integers satisfying 1a31 1 \leq a \leq 31, 1b311 \leq b \leq 31 such that the polynomial P(x)=x3ax2+a2b3x+9a2b2P(x)=x^3-ax^2+a^2b^3x+9a^2b^2 has roots rr, ss, and tt.

Given that there exists a positive integer kk such that (r+s)(s+t)(r+t)=k2(r+s)(s+t)(r+t)=k^2, compute the maximum possible value of abab.

This problem is posed by Samir K.

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