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A new optimal distributed linear averaging (ODLA) problem is presented in this paper. This problem is motivated from the distributed averaging problem which arises in the context of distributed algorithms in computer science and coordination of groups of autonomous agents in engineering. The aim of the ODLA problem is to compute the average of the initial values at nodes of a graph through an optimal distributed algorithm in which the nodes in the graph can only communicate with their neighbors. Optimality is given by a minimization problem of a quadratic cost functional under infinite horizon. We show that this problem has a very close relationship with the notion of semistability. By developing new necessary and sufficient conditions for semistability of linear discrete-time systems, we convert the original ODLA problem into two equivalent optimization problems. These two problems are convex optimization problems and can be solved by using some numerical techniques.