Statement

There are N+2 friends, indexed from 0 to N+1, who gathered to play a game of Rummy. After playing many rounds, each friend obtained some score. The score of friend i is an integer P_i (in Rummy, scores can be negative). Friends 0 and N+1 obtained a non-negative score. This made the other N friends jealous, so they devised the following maneuver in order to make every score non-negative: (1) Friend i (1 < i < N) gives one point to each of they two neighbours. In other words, after friend i applies the maneuver, their score decreases by two (P_i = P_i - 2), and the scores of friends i-1 and i+1 increase by one (P_i-1 = P_i-1 + 1, P_i+1 = P_i+1 + 1). Note that friends 0 and N+1 are not allowed to apply the maneuver. Each friend can apply the maneuver as many times as they want, even if their score is negative, or becomes negative after the maneuver. Determine whether every friend can have a non-negative score (all at the same time) after applying some number of maneuvers.