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A spatial monitoring query (SMQ) retrieves the set of mobile nodes that satisfy some spatial constraints, and provides real-time updates whenever this set of nodes changes. Efficient processing of such queries is essential to moving objects database management. Existing techniques rely on one or more central servers for query management and assume each mobile node can communicate with some server directly. These limitations prevent them from being used in application scenarios where no such server exists. This paper assumes a mobile peer-to-peer system where mobile nodes are the only computing devices, and investigates the challenges of allowing mobile nodes to collaborate in query processing. We present a cost-effective technique to process a primitive type of SMQs and then show that other types of queries can be converted into the primitive type of queries. As such, different types of queries can now be supported within a common platform and without relying on any stationary server. We also evaluate, through both mathematical analysis and simulation, the performance of the proposed platform in terms of mobile communication costs incurred in query processing.