KURSCHAK 2013 Problem 2

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 $[P_2]>4$ .