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**particlejohn** Let $\displaystyle T $ be an acute triangle. Let $\displaystyle R $ and $\displaystyle S$ be inscribed rectangles in $\displaystyle T $. Also, let $\displaystyle A(X) $ be the area of a polygon $\displaystyle X $. Does $\displaystyle \frac{A(R)+A(S)}{A(T)} $ have a maximum? If so, what is it? $\displaystyle R $ and $\displaystyle S $ range over all rectangles and $\displaystyle T $ ranges over all triangles.

So they probably gave $\displaystyle T $ as an acute triangle for some reason. I am not sure why we need both $\displaystyle R $ and $\displaystyle S $, since they are both rectangles. If you just let $\displaystyle R $ range over all rectangles, wouldn't that "cover" $\displaystyle S $? (e.g. we could instead consider $\displaystyle \frac{A(R)}{A(T)} $)? Now $\displaystyle A(T) $ is the area of all the triangles inside $\displaystyle T $ right?

Any ideas? Use any derivative tests?