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In current peer-to-peer file sharing networks, a large number of peers with heterogeneous connections simultaneously seek to download resources, e.g., files or file fragments, from a common seed at the time these resources become available, which incurs high download delays on the different peers. Unlike existing literature which mainly focused on cooperative strategies for data exchange between different peers after all the peers have already acquired their resources, in this paper, we study the cooperation possibilities among a number of peers seeking to download, concurrently, a number of resources at the time the availability of the resources is initially announced at a common seed. We model the problem as a coalitional game in partition form and we propose an algorithm for coalition formation among the peers. The proposed algorithm enables the peers to take autonomous decisions to join or leave a coalition while minimizing their average download delay. We show that, by using the proposed algorithm, a Nash-stable partition composed of coalitions of peers is formed. Within every coalition, the peers distribute their download requests between the seed and the cooperating partners in a way to minimize the total average delay incurred on the coalition. Analytically, we study the 2-peer scenario and derive the optimal download request distribution policies. Simulation results show that, using the proposed coalition formation game, the peers can improve their average download delay per peer of up to 99.6% compared to the non-cooperative approach for the case with N = 15 peers.