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max \frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2} if bc-ca-ab=1

Source: Mathcenter Contest / Oly - Thai Forum 2012 (R1) p6 sl-14 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 13, 2022

algebrainequalitiesmax

Problem Statement

For a real number a,b,c>0 where

bc-ca-ab=1

find the maximum value of

P=\frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2}

and find out when that holds .
(PP-nine)