I always prove in the contests that can only submit once(CNOI mode)

I usually prove it if it's easy and the pernalty affects

I usually prove-by-ac in an IOI-mode contest and the method is easy-coding.

I don't bother with formality and rigor, but I always try to at least justify any observations I make.

No most of the time. We usually come up with examples* to verify the hypotheses. And rely on gut feeling too. Sounds iffy but solving competitive programming problems doesn't require much rigor. Sure, you should test your submission carefully and make sure not to make silly mistakes during a contest but mathematical proofs sound way over the top.

I have nothing against proofs, though. Proofs are awesome. They're just not that useful during a contest.

* Your brain isn't a good random number generator. It is rather biased. Write a computer program to generate test cases instead. You can work them out by hand, however. A dozen cases or so and you can be very confident that your algorithm is correct.

Also I second what Shisuko said.

Not exactly prove, but I try to convince myself about some observations I made using contradictions and inductions.

I trust in my luck :v

I won't ever use an observation I cannot justify.