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A constant independent of n!

Source: India TST 2001 Day 3 Problem 3

January 31, 2015
geometryinradiusgeometry proposed

Problem Statement

Points B=B1,B2,⋯ ,Bn,Bn+1=CB = B_1 , B_2, \cdots , B_n , B_{n+1} = C are chosen on side BCBC of a triangle ABCABC in that order. Let rjr_j be the inradius of triangle ABjBj+1AB_jB_{j+1} for j=1,⋯ ,nj = 1, \cdots, n , and rr be the inradius of △ABC\triangle ABC. Show that there is a constant λ\lambda independent of nn such that : (λ−r1)(λ−r2)⋯(λ−rn)=λn−1(λ−r)(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)
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