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Maximum value of Integral on subset of plane

Source: 2018 South Korea USCM P3

August 14, 2020
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Problem Statement

Φ\Phi is a function defined on collection of bounded measurable subsets of R\mathbb{R} defined as Φ(S)=∬S(1−5x2+4xy−5y2)dxdy\Phi(S) = \iint_S (1-5x^2 + 4xy-5y^2 ) dx dy Find the maximum value of Φ\Phi.
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