IMO 1982 LL AUS2

Origin: AUS

Problem

Given a finite number of angular regions A1, . . . , Ak in a plane, each Ai being bounded by two half-lines meeting at a vertex and provided with a + or −sign, we assign to each point P of the plane and not on a bounding half-line the number k −l, where k is the number of + regions and l the number of −regions that contain P. (Note that the boundary of Ai does not belong to Ai.) For instance, in the figure we have two + regions QAP and RCQ, and one −region RBP. Every point in- side \triangleABC receives the number




A A A AA  A B C P Q R + − + +1, while every point not inside \triangleABC and not on a boundary halfline the number 0. We say that the interior of \triangleABC is represented as a sum of the signed angular regions QAP, RBP, and RCQ. (a) Show how to represent the interior of any convex planar polygon as a sum of signed angular regions. (b) Show how to represent the interior of a tetrahedron as a sum of signed solid angular regions, that is, regions bounded by three planes inter- secting at a vertex and provided with a + or −sign.