Exactly. The comment at the end "Now imagine there are hundreds of other entrants, even further decreasing your odds" makes no sense. Those hundreds (actually, more like you said: tens of thousands) of other entrants make the negative effect of inviting your own friends negligible.

Example: let's assume there 10,000 entrants including you, maintain the assumption that "a new person will sign up each day based on your initial invitation", and give it 30 days. Therefore, you only get 1 extra entry and 30 of your extended friends have signed up (each getting an extra entry except the last). Your chances have gone from:

1/10,000

to

2/(10,000+30+29)

You've increased your odds by 98.8%! And you could do even better by waiting until right before the contest deadline and inviting as many people as possible.

The break even point is ridiculously low too: only 60 entrants under these assumptions. So even with just "hundreds of other entrants", you should share.

Keep looking for that science, bro.