Consider the closed polygonal discs $P_1$, $P_2$, $P_3$ with the property that for any three points $A\in P_1$, $B\in P_2$, $C\in P_3$, we have $[\triangle ABC]\le 1$. (Here $[X]$ denotes the area of polygon $X$.)
- (a) Prove that $\min\{[P_1],[P_2],[P_3]\}<4$.
- (b) Give an example of polygons $P_1,P_2,P_3$ with the above property such that $[P_1]>4$ and $[P_2]>4$.
How to do this problem? Can someone help me? Thanks. This is a problem from Hungarian Math Olympiad.