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unbounded sequence of positive integers by Manolescu

Source: Romanian IMO Team Selection Test TST 1999, problem 8

September 24, 2005
inductionalgebra proposedalgebra

Problem Statement

Let aa be a positive real number and {xn}n≥1\{x_n\}_{n\geq 1} a sequence of real numbers such that x1=ax_1=a and xn+1≥(n+2)xn−∑k=1n−1kxk, ∀ n≥1. x_{n+1} \geq (n+2)x_n - \sum^{n-1}_{k=1}kx_k, \ \forall \ n\geq 1. Prove that there exists a positive integer nn such that xn>1999!x_n > 1999!. Ciprian Manolescu
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