At first glance it doesn't appear to be server side (I won't rule it out of course), I have tested on 2 different PCs and net connections and stay logged in. Some things to check - what browser are you using, what operating system? Do you have cookies enabled in your browser? Have any security updates been done recently, do you go through a company proxy server etc. If you need me to explain any of these I would be happy to help, just need to identify the environment the site is being viewed from as when it comes to things such as login sessions there are many things that can distrupt it.
I use Firefox (and the latest windows I think?). I did clear my history now and made sure that cookies are enabled.
Now it works when I move from page 1 to page 2, but when I "refresh" on my browser bar I log out again.