Question 64
Petya chose 100 pairwise distinct positive real numbers, each less than $1$, and arranged these numbers on a circle. Then he performs the following moves. By one move he takes some three consecutive numbers $a, b, c$ (in this order) and replaces the middle number $b$ by $a - b + c$. Find the greatest possible $k$ such that Petya can choose initial numbers and perform several moves so that in the resulting arrangement there are $k$ integers.