IMO Shortlist 1970 problem 5


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Let M be an interior point of the tetrahedron ABCD. Prove that \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) + \stackrel{\longrightarrow }{MB} \text{vol}(MACD) + \stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0\text{.}
(\text{vol}(PQRS) denotes the volume of the tetrahedron PQRS).
Source: Međunarodna matematička olimpijada, shortlist 1970