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Omni-MATH / Assume that (a_(n))_(n≥1) is an increasing sequence of positive real numbers such that lima_(n)/n=0. Must there exist infinitely many positive integers n such that …

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problem

Assume that (an)n≥1(a_n)_{n\geq 1} is an increasing sequence of positive real numbers such that liman/n=0\lim a_n/n=0. Must there exist infinitely many positive integers nn such that an−i+an+i<2ana_{n-i}+a_{n+i}<2a_n for i=1,2,…,n−1i=1,2,\ldots,n-1?
Plain-text mathematical notation (without MathML)
Assume that (a_(n))_(n≥1) is an increasing sequence of positive real numbers such that lima_(n)/n=0.  Must there exist infinitely many positive integers n such that a_(n−i)+a_(n+i)<2a_(n) for i=1,2,…,n−1?
Original LaTeX notation
Assume that $(a_n)_{n\geq 1}$ is an increasing sequence of positive real numbers such that $\lim a_n/n=0$.  Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\ldots,n-1$?

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Source and history

task_id
2d5146fd-17ce-5ce3-89b1-74158321aa2a
task_key
test--2d5146fd-17ce-5ce3-89b1-74158321aa2a
task_revision_id
1
config / split
/ test

Official source

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