Question 172
We say that a sequence $a_1, \dots, a_n$ of real numbers is decreasingly increasing if for all $1 \lt i \lt n$ we have $0 \lt a_{i+1} - a_i \lt a_i - a_{i-1}$. Find, for every positive integer $m$, the smallest positive integer $k$ for which there exists a decreasingly increasing sequence of length $k$ such that 1 can be written in at least $m$ different ways as the difference of two elements $a_i$ and $a_j$ of the sequence.