Theoretical problem about coinflips

[AleoMagus]

Maybe easy to answer, and I certainly think I know the answer to this, but I'd like to see a definitive closure to the following question:

Assuming Independent chip model is accurate, opponents are equal in ability, and blind sizes are negligible to the problem, is there any situation where a known coinflip should be taken when an opponent pushes all in before you? Also assume that no matter how many players are at the table, showdown will only involve you and the lone pusher.

Put another way, if I know for CERTAIN that my odds of beating an opponent in a showdown are exactly 50/50, and this opponent pushes all in ahead of me, are there any conditions under which I can call profitably.

So just pull out the ICM calculator and find me one example.

Some more complicated questions to answer if you are feeling really ambitious:

If there are circumstances where you can call an opponent profitably with a true coinflip, what is special about those circumstances in general that makes this so?

Conversely, if there are no circumstances under which this is the case, I'd like to see a proof.

Regards
Brad S

As a corrollary to this problem, does the answer to this make sense given what we know about the relative value of chips in different sized stacks?

Ie - common tourney knowledge often suggests that calling with a suspected coinflip is wise if we have a huge stack in comparison. This is becasue the small stack's chips are worth more than ours and we may actually have pot odds to do so when considering the 'extra' value of shorty's chips.