1985 IMO 第 5 题

(ROM 1) Let $D$ be the interior of the circle $C$ and let $A \in C$. Show that the function $f: D \rightarrow \mathbb{R}, f(M)=\frac{|M A|}{\left|M M^{\prime}\right|}$, where $M^{\prime}=(A M \cap C$, is strictly convex; i.e., $f(P)<\frac{f\left(M_{1}\right)+f\left(M_{2}\right)}{2}, \forall M_{1}, M_{2} \in D, M_{1} \neq M_{2}$, where $P$ is the midpoint of the segment $M_{1} M_{2}$.

(ROM 1) 令$D$ 为圆$C$ 的内部，并令$A \in C$。证明函数 $f: D \rightarrow \mathbb{R}, f(M)=\frac{|M A|}{\left|M M^{\prime}\right|}$，其中 $M^{\prime}=(A M \cap C$, 是严格凸的；即 $f(P)<\frac{f\left(M_{1}\right)+f\left(M_{2}\right)}{2}, \forall M_{1}, M_{2} \in D, M_{1} \neq M_{2}$，其中 $P$ 是线段 $M_{1} M_{2}$ 的中点。