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An arbitrary pentagon (which is convex) is given to you, and it has a perimeter of $k$. Determine the i) maximum area, rigorously, in terms of $k$. ii) maximum INTEGRAL area, in terms of $k$.

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    *Among all $n$-gons inscribed in a circle, which one has largest area and which one has largest perimeter? The regular polygon is the expected answer, and this turns out to be the case.* Maxima and minima without calculus by Ivan Morton Niven Volume 6, 6.1 Introduction.2011-08-27

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