MathDB
Nice sequence bound

Source: Bulgaria MO Regional round 2024, 12.2

February 13, 2024
algebrainequalitiesanalysisSequence

Problem Statement

Let NN be a positive integer. The sequence x1,x2,…x_1, x_2, \ldots of non-negative reals is defined by xn2=∑i=1n−1xixn−ix_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}} for all positive integers n>Nn>N. Show that there exists a constant c>0c>0, such that xn≤n2+cx_n \leq \frac{n} {2}+c for all positive integers nn.
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