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Sequence - Proof of 0-value term

Source: Canadian Mathematical Olympiad - 1995 - Problem 5.

May 9, 2011
parameterizationtrigonometryalgebra unsolvedalgebra

Problem Statement

uu is a real parameter such that 0<u<10<u<1. For 0≤x≤u0\le x \le u, f(x)=0f(x)=0. For u≤x≤nu\le x \le n, f(x)=1−(ux+(1−u)(1−x))2f(x)=1-\left(\sqrt{ux}+\sqrt{(1-u)(1-x)}\right)^2. The sequence {un}\{u_n\} is define recursively as follows: u1=f(1)u_1=f(1) and un=f(un−1)u_n=f(u_{n-1}) ∀n∈N,n≠1\forall n\in \mathbb{N}, n\neq 1. Show that there exists a positive integer kk for which uk=0u_k=0.
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