Normally new users are not allowed to up-vote until they gain a bit of reputation by receiving up-votes or accepts from the community of a site. 
But this has a weak point that is being exploited:

> A number of new users post, answer, and accept each others' 
off-topic questions before we can close them (in less than 5 min). 
Accepting each others' answers gains them enough reputation to 
start up-vote each others' posts.

We are closing them and the moderators clean up afterwards but 
this keeps happening over and over again and becoming frustrating 
to the extent that I think a systematic solution might be required.
Is there a system to prevent this kind of circumvention of 
the reputation requirements using accepts by new users?

One possible solution might be to change the system so
accepts from users who do not have enough reputation to up-vote 
do not give any reputation (at least until the answer gets an up-vote).