Hotter and Fresher

Algebra Level 5

\[\large P=3^{x+y-4}+(x+y+1)\cdot 2^{7-x-y}-3(x^2+y^2)\] Given that \(x\) and \(y\) are real numbers satisfying \[x+y+1=2(\sqrt{x-2}+\sqrt{y+3}). \]

If the maximum value of \(P\) can be expressed as \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).


This problem is extracted from the 2016 Vietnamese University Entrance Examination which just took place a few hours ago. Solutions and discussions are always welcome!
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