Question 113
$a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ are $2n$ positive numbers. We know that all the $a_i$'s, $1 \le i \le n$ are not equal, and that they can be separated into two partitions of equal sum. These two properties hold for the $b_i$'s, $1 \le i \le n$, as well. Prove that there exists a simple $2n$-gon with sides parallel to the coordinate axes such that the lengths of its horizontal edges are equal to $a_i$'s and the lengths of its vertical edges are equal to $b_i$'s (a simple polygon is one that does not cross itself).