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sequences of polygons, vertices are midpoints of sides, convergence

Source: SEEMOUS 2008 P2

June 17, 2021
Sequencesgeometryconvex polygon

Problem Statement

Let P0,P1,P2,…P_0,P_1,P_2,\ldots be a sequence of convex polygons such that, for each k≥0k\ge0, the vertices of Pk+1P_{k+1} are the midpoints of all sides of PkP_k. Prove that there exists a unique point lying inside all these polygons.
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