Problem 1

In a convex quadrilateral of area

32

, the sum of the lengths of two opposite sides and one diagonal is

16

. Determine all possible lengths of the other diagonal.

Step 4 of 7: Optimize the area bound

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In plain words

The given area is exactly the global maximum of the bound.

32\le\frac{BD(16-BD)}2\le32

Detailed analysis

Using the sum condition, $32=\mathrm{Area}\le\frac{BD(16-BD)}2$ . The quadratic $f(x)=x(16-x)/2$ has maximum $32$ at $x=8$ , hence both inequalities are equalities and $BD=8$ .