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IMC 2011 Day 2, Problem 5

Source:

August 1, 2011
inductiongeometryperpendicular bisectorIMCcollege contests

Problem Statement

Let F=A0A1...AnF=A_0A_1...A_n be a convex polygon in the plane. Define for all 1≤k≤n−11 \leq k \leq n-1 the operation fkf_k which replaces FF with a new polygon fk(F)=A0A1..Ak−1Ak′Ak+1...Anf_k(F)=A_0A_1..A_{k-1}A_k^\prime A_{k+1}...A_n where Ak′A_k^\prime is the symmetric of AkA_k with respect to the perpendicular bisector of Ak−1Ak+1A_{k-1}A_{k+1}. Prove that (f1∘f2∘f3∘...∘fn−1)n(F)=F(f_1\circ f_2 \circ f_3 \circ...\circ f_{n-1})^n(F)=F.
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