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x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0

Source: 2015 Saudi Arabia BMO TST I p1

July 24, 2020
Sequencerecurrence relation

Problem Statement

Prove that for any integer n≥2n \ge 2, there exists a unique finite sequence x0,x1,...,xnx_0, x_1,..., x_n of real numbers which satisfies x0=xn=0x_0 = x_n = 0 and xi+1−8xi3−4xi+3xi−1+1=0x_{i+1} - 8x_i^3 -4x_i + 3x_{i-1} + 1 = 0 for all i=1,2,...,n−1i = 1,2,...,n - 1. Prove moreover that ∣xi∣≤12 |x_i| \le \frac12 for all i=1,2,...,n−1i = 1,2,...,n - 1.
Nguyễn Duy Thái Sơn
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