We have N objects with weights $1, 2,\cdots , N$ grams. We wish to choose two or more of these objects so that the total weight of the chosen objects is equal to average weight of the remaining objects. Prove that (a) (2 points) if $N + 1$ is a perfect square, then the task is possible; (b) (6 points) if the task is possible, then $N + 1$ is a perfect square.