2001 Putnam B6

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 $i=1,2,\ldots,n-1$?

假设 $(a_n)_{n\geq 1}$ 是一个递增序列

正实数使得

$\lim a_n/n=0$。是否一定存在无穷多个正整数

$n$ 使得 $a_{n-i}+a_{n+i}<2a_n$ 对于 $i=1,2,\ldots,n-1$？