I'm pretty sure this is an obvious call, but I wasn't happy about it....

Satellite tournament, winner gets a $1500 seat to wsop, 2nd place gets $100. 9 hours in, last 2 players:

Hero, BTN: 90,000
Villain, BB: 110,000

blinds are 2000/4000, 500 ante

Hero has AQo, raises to 12,000 (has been standard preflop raise)
villain shoves

This is about 20 hands into heads-up play. Hero started with only about 40,000, but has battled back with a ton of mostly unanswered preflop aggression and one decent pot. Villain is a very solid regular in the league, but we haven't seen him play heads-up before. So far, he seems pretty timid and tight, not adjusting to heads-up properly. If I fold, I'm thinking I have a pretty strong advantage in heads-up play, but that's not based on a lot of history. The overbet here seems a bit off; neither of us has shoved like this yet.

Call?

i think its a quick call.
he probably dont have AA, KK, QQ that have you crushed. his big overbet inticates he want this pot here and now,
more likely he have middle pair like 9 9, T T, and then its a classic race, or maybe even AJ, AT, KQ, and then you are in a good position.

thing is, I'm not very happy to take a race right now. I'm feeling good about heads up so far, and would much rather play with the equal stacks than flip. If I thought there was a strong likelihood I had him dominated, sure. I actually couldn't pare his range down much at all in my head.

If I fold I only have ~40% of the chips, though. That's a substantial disadvantage and I think I have him dominated often enough to be 55% or more if I call.

if he shows me 55, I'm folding. I think I have better than 45% overall with 40% of the chips.

If you feel you can outplay him and he's letting you see a bunch of boards, then I fold.

Against a weaker player I want to nibble at his chip stack, not put the whole tourney into a race.

Against a stronger player, I want to race as I know I'm not likely to beat him post flop.

In fact I remember hearing that if you were an amateur player going up against a pro heads up, the smartest thing you can do is open-shove regularly, as it takes away his ability to play strong on later streets.

I would apply the same principle here.

I think calling is clearly the right play here. I find a lot of inexperienced heads up players make this play with 22+ or any Ax. Obviously 22+ wouldn't be ideal to go up against, but you need to consider a good portion of Ax as part of his range...

Now I understand your argument that you think you can outplay him the rest of the way, but inevitably in a heads up format you are going to get into a spot where the money is going in when you're 60/40 (A9 vs. KQ or something). So I think that there is no reason why your opponent wouldn't continue to make this play and keep putting you into this tough spot when he has 22+ or Ax, so you should make your stand sooner than later....

Why if you are seeing alot of flops and turns would you ever have to get your money in a 60/40 spot?

Why flip a coin when you can likely get it in when you are at least a 70% favourite?

good points so far, I'm pleasantly surprised people are taking the question seriously. Maybe I wasn't crazy to pause and think about this.
Thing is, this shove was a departure from his earlier play. It may be a response to the way heads-up has been going (pretty one-sided), and he's now saying he's going to force me to flip for all the chips rather than bleed his big stack away a little at at time. If that's the case, right now I've got the best flipping hand I'm likely to see, and will seriously regret having folded it.

...Or, he's been waiting for a top-5% hand to crush the maniac.

[fwiw, I've been taking pots down preflop, mostly, not seeing flops. only 3 hands have gone to the flop. I won one at showdown, had to fold one, and won the 3rd with a bold cbet]

If you fold, your M is just over 10, and you are headed towards shove/fold play as the blinds raise. If you are a 45% dog to a middle pair, then the $EV of the call is $730.

If you think that you have a 5% edge on him, folding leaves you with 67,000 chips and him with 133,000. If you add your edge to a standard chip split, you are about 40% to win after the fold.

You would need to believe you had about a 12% edge on him in skill--or better--to make this a good fold.

folding here will leave you with less then 20 BB, with villain stack is 50% bigger then yours.
if you think you advantage against villain is big- fold might be the good idea.
however, with blinds so big, your advantage probably is not as big as you think. next hand- you raise with KQ, and he put you all in (again). can you realy fold again?!? you will have to take a stand sometime, and AQ is a realy good spot..