From the 2015 Moscow Mathematical Olympiad:

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> The numbers $1$ to $1000$ are arranged on a circle such that each number divides the sum of its two neighbors. Suppose that the number $k$ has two odd neighbors. What's the parity of $k$ ?
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I haven't been able to answer the question just yet. 

Here are a few things I noticed. The constraint "$x$ divides its two neighbors $a$ and $b$" gets tighter when $x$ grows. For example, if $x=1000$, we must have $1000=a+b$. If $x=999$, we have either $999=a+b$ or $1998=a+b$ (in that case $(a,b)\in \{(1000,998),(998,1000)\}$).

Nevertheless, I can't derive anything useful from $k| a+b$ with $a,b$ being odd numbers.