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Bounded sequences for choice of first term of the sequence

Source: Simon Marais Mathematics Competition 2023 Paper A Problem 4

October 14, 2023
algebraSequences

Problem Statement

Let x0,x1,x2…x_0, x_1, x_2 \dots be a sequence of positive real numbers such that for all n≥0n \geq 0, xn+1=(n2+1)xn2xn3+n2x_{n+1} = \dfrac{(n^2+1)x_n^2}{x_n^3+n^2} For which values of x0x_0 is this sequence bounded?
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