12.2

Part of 2024 Bulgaria MO Regional Round

Problems(1)

Nice sequence bound

Source: Bulgaria MO Regional round 2024, 12.2

N

be a positive integer. The sequence

x_1, x_2, \ldots

of non-negative reals is defined by

x_n^2=\sum_{i=1}^{n-1} \sqrt{x_ix_{n-i}}

for all positive integers

n>N

. Show that there exists a constant

c>0

x_n \leq \frac{n} {2}+c

for all positive integers

n

algebrainequalitiesanalysisSequence