IMO Shortlist 1970 problem 5
Dodao/la:
arhiva2. travnja 2012. Let
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
be an interior point of the tetrahedron
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
. 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{.}](/media/m/6/9/1/691955b9b882529ed4473ef0fbfd7236.png)
(
![\text{vol}(PQRS)](/media/m/4/1/6/416a5c22fc1e4cb45e2e325bbf1565c5.png)
denotes the volume of the tetrahedron
![PQRS](/media/m/6/6/3/663b967cae96529d4dd7efbd0009ec54.png)
).
%V0
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 $PQRS$).
Izvor: Međunarodna matematička olimpijada, shortlist 1970