How can I replicate MySQL data from many servers to one? All over the internet I can find manuals on how to replicate databases from master(s) to slaves. 

What I need is to continuously keep a copy of table from 5 different servers in sync.

We have one application running at 5 locations. We need to have a copy of the system_log table from every one of those 5 servers always in sync on one new server at our 'headquarters'.

What is the best way to achieve this without using custom scripts?
Sorry.  If you are trying to have many Masters and one Slave -- that is not supported.  (Some third party addons may have such.)  It is better to have the 5 servers all write to the 'headquarters' Master.  Network latencies are probably not a serious problem.