1970 IMO 第 5 题

(CZS 3) Let $M$ be an interior point of the tetrahedron $A B C D$. Prove that $$\begin{aligned} & \overrightarrow{M A} \operatorname{vol}(M B C D)+\overrightarrow{M B} \operatorname{vol}(M A C D) \\ & \quad+\overrightarrow{M C} \operatorname{vol}(M A B D)+\overrightarrow{M D} \operatorname{vol}(M A B C)=0 \end{aligned}$$ ( $\operatorname{vol}(P Q R S)$ denotes the volume of the tetrahedron $P Q R S)$.

(CZS 3) 令$M$ 为四面体$A B C D$ 的内点。证明 $$\begin{aligned} & \overrightarrow{M A} \operatorname{vol}(M B C D)+\overrightarrow{M B} \operatorname{vol}(M A C D) \\ & \quad+\overrightarrow{M C} \operatorname{vol}(M A B D)+\overrightarrow{M D} \operatorname{vol}(M A B C)=0 \end{aligned}$$ ( $\operatorname{vol}(P Q R S)$ 表示四面体 $P Q R S)$ 的体积。