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4 conditions' mapping from set of polygons

Source: North Macedonian Mathematical Olympiad 1996 p2

February 12, 2020
functionpolygonrectangle

Problem Statement

Let PP be the set of all polygons in the plane and let M:P→RM : P \to R be a mapping that satisfies: (i) M(P)≥0M(P) \ge 0 for each polygon PP, (ii) M(P)=x2M(P) = x^2 if PP is an equilateral triangle of side xx, (iii) If a polygon PP is partitioned into polygons SS and TT, then M(P)=M(S)+M(T)M(P) = M(S)+ M(T), (iv) If polygons PP and TT are congruent, then M(P)=M(T)M(P) = M(T ). Determine M(P)M(P) if PP is a rectangle with edges xx and yy.
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