I am using KM 1.02. There are several web sites that I use regularly that have added security questions to their login process. Apparently the web site checks to see if this computer has logged in before, and, if not, requires you to answer a security question before proceeding.

I use Tools/Privacy/Settings and check "Clear on Shutdown" for all options, so I assume I am deleting whatever the web site puts on my computer when I close KM.

Are they doing this trough cookies? Is there some way I can keep something on my computer so that these web sites will know that I have logged in before.

Probably cookies ( I read that there is another theoretical way via cache data but I have never hear that it was used - and have no idea how it can function in practise )

Try:
Set that cookies are not cleared privacy plugin.
Set to reject all cookies via Preferences > Privacy
White List the sites that You want to set cookies so that they can recoginze You.

That is my only idea.

will k-meleon delete ALL cookies if delete cookies on exit is enabled?