Remove old solutions from the Start page.

There is a KB article that explains exactly how to do this:

http://kb.k2workflow.com/Articles/KB000013.aspx

in short, K2.net 2003 Studio stores its Recent Solution List in the registry and specifically in the following entries:
HKEY_CURRENT_USERSOFTWARESourceCodeK2.net StudioRecent File List
HKEY_CURRENT_USERSOFTWARESourceCodeK2.net StudioRecentSolutionPaths

you can manually remove the enries using regedit (not recommended due to the sensitive nature of editing registry entries). The KB article has a .vbs script sample you can use to clear the recent solutions list.