Note that the only number that can be placed adjacent to $50$ is $1$. So $50$ must be the first or last number. WLOG it's the first. We can prove by induction that the $(2i)^{th}$ and $(2i+1)^{th}$ number in the sequence must be $i$ and $i+50$ for all $i\in \{ 1,2,...,48\}$. And finally, the last number is $49$. Similarly, if $50$ is the last number, we will get the sequence of this case in backward order. So, there are exactly $2$ sequences satisfied the requirements.

I think this is the Cono Sur Olympiad 1998 P1. The 1997 P1 can be found here: http://www.oma.org.ar/enunciados/con8.htm