Question 4
Let $n \geqslant 2$ and let $a_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{n}$ be strictly positive real numbers. The $a_{i}$ are assumed to be pairwise distinct, and so are the $b_{i}$. Assume that it is possible to split the $a_{i}$ into two subsets with the same sum, and likewise for the $b_{i}$.
Show that there exists a simple $2n$-gon with exactly $n$ horizontal sides and $n$ vertical sides, such that the lengths of the horizontal sides are exactly the $a_{i}$ and the lengths of the vertical sides are exactly the $b_{i}$.
Show that there exists a simple $2n$-gon with exactly $n$ horizontal sides and $n$ vertical sides, such that the lengths of the horizontal sides are exactly the $a_{i}$ and the lengths of the vertical sides are exactly the $b_{i}$.