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I want to make sure I understand the rules for Paired Opportunists when combined with Greater Trip and Vicious Stomp.
We have two characters, Lean and Mean, who both threaten an helpless Victim. Lean fights unarmed and has the feats Greater Trip, Vicious Stomp, and Combat Reflexes. Mean has Solo Tactics, Paired Opportunists, and Combat Reflexes.
On Lean's turn, he successfully trips Victim.
1) Because of Greater Trip, Victim provokes attacks of opportunity from both Lean and Mean. These AoO occur before Victim becomes prone, and thus Victim does not take an AC penalty.
2) Because of Vicious Stomp, Victim provokes an unarmed AoO specifically from Lean. The prone penalty to AC applies.
3) Because Lean took an AoO that Mean did not also get, Paired Opportunists gives Mean an AoO against Victim, with the prone penalty to AC.
4) On Victim's turn, he stands up, provoking once from both Lean and Mean. Since these resolve before Victim reaches his feat, prone penalty to AC applies.
So Lean and Mean each get 3 AoO from the one trip, and the second two both attack AC-4. Is that correct?
And the follow-up, if only Lean threatens Victim at the start of his trip, but he uses Ki Throw to land Victim next to Mean, does Mean still get the second two AoO (but not the first, since Mean doesn't threaten at the start of the trip)? Note: I am aware that Ki Throw itself doesn't provoke AoO, but believe the Vicious Stomp / Paired Opportunists should still key off.

Anachrony |

My question is when you get your unarmed vicious stomp AoO, and your partner gets an AoO from Paired Opportunists, does the partner's AoO also have to be unarmed?
My interpretation is no, they can do a regular AoO of their choice. Because they're not actually getting it from Vicious Stomp themselves, they're getting a generic AoO from Paired Opportunists triggered by another AoO but not subject to the same conditions as that AoO, "even if the situation or an ability would normally deny you the attack of opportunity". I think it works in terms of RAW, and in terms of RAI, my understanding of the concept is that your AoO distracts them and creates an opening.

Elbedor |

The AoOs all enjoy the AC penalty. Greater Trip's AoO takes place after the target has been "successfully tripped" (i.e. knocked prone).
Of course this has been a point of discussion quite a bit on this board. ;) But being "tripped" is defined throughout the rules as the activity of being knocked prone and not simply "a trip attack" which is related, but something entirely different.
For reference, consider the Trip feature of weapons or the Trip ability of creatures. They mention "not being tripped in return" which is referencing the act of being "knocked prone" as no return "trip attack" is ever made on a failed attempt.

Mucronis |
The AoOs all enjoy the AC penalty. Greater Trip's AoO takes place after the target has been "successfully tripped" (i.e. knocked prone).
Of course this has been a point of discussion quite a bit on this board. ;) But being "tripped" is defined throughout the rules as the activity of being knocked prone and not simply "a trip attack" which is related, but something entirely different.
For reference, consider the Trip feature of weapons or the Trip ability of creatures. They mention "not being tripped in return" which is referencing the act of being "knocked prone" as no return "trip attack" is ever made on a failed attempt.
Combat Maneuver Bonus: Each character and creature has a Combat Maneuver Bonus (or CMB) that represents its skill at performing combat maneuvers. A creature's CMB is determined using the following formula:
CMB = Base attack bonus + Strength modifier + special size modifier
Creatures that are size Tiny or smaller use their Dexterity modifier in place of their Strength modifier to determine their CMB. The special size modifier for a creature's Combat Maneuver Bonus is as follows: Fine –8, Diminutive –4, Tiny –2, Small –1, Medium +0, Large +1, Huge +2, Gargantuan +4, Colossal +8. Some feats and abilities grant a bonus to your CMB when performing specific maneuvers.
Performing a Combat Maneuver: When performing a combat maneuver, you must use an action appropriate to the maneuver you are attempting to perform. While many combat maneuvers can be performed as part of an attack action, full-attack action, or attack of opportunity (in place of a melee attack), others require a specific action. Unless otherwise noted, performing a combat maneuver provokes an attack of opportunity from the target of the maneuver. If you are hit by the target, you take the damage normally and apply that amount as a penalty to the attack roll to perform the maneuver. If your target is immobilized, unconscious, or otherwise incapacitated, your maneuver automatically succeeds (treat as if you rolled a natural 20 on the attack roll). If your target is stunned, you receive a +4 bonus on your attack roll to perform a combat maneuver against it.
When you attempt to perform a combat maneuver, make an attack roll and add your CMB in place of your normal attack bonus. Add any bonuses you currently have on attack rolls due to spells, feats, and other effects. These bonuses must be applicable to the weapon or attack used to perform the maneuver. The DC of this maneuver is your target's Combat Maneuver Defense. Combat maneuvers are attack rolls, so you must roll for concealment and take any other penalties that would normally apply to an attack roll.
Combat Maneuver Defense: Each character and creature has a Combat Maneuver Defense (or CMD) that represents its ability to resist combat maneuvers. A creature's CMD is determined using the following formula:
CMD = 10 + Base attack bonus + Strength modifier + Dexterity modifier + special size modifier
The special size modifier for a creature's Combat Maneuver Defense is as follows: Fine –8, Diminutive –4, Tiny –2, Small –1, Medium +0, Large +1, Huge +2, Gargantuan +4, Colossal +8. Some feats and abilities grant a bonus to your CMD when resisting specific maneuvers. A creature can also add any circumstance, deflection, dodge, insight, luck, morale, profane, and sacred bonuses to AC to its CMD. Any penalties to a creature's AC also apply to its CMD. A flat-footed creature does not add its Dexterity bonus to its CMD.
Determine Success: If your attack roll equals or exceeds the CMD of the target, your maneuver is a success and has the listed effect. Some maneuvers, such as bull rush, have varying levels of success depending on how much your attack roll exceeds the target's CMD. Rolling a natural 20 while attempting a combat maneuver is always a success (except when attempting to escape from bonds), while rolling a natural 1 is always a failure.
Being "tripped" is NOT defined throughout the rules as knocked prone, the bolded part in the spoiler defines it as roll of the die, but let's not get into that discussion here.
Trip
You can attempt to trip your opponent in place of a melee attack. You can only trip an opponent who is no more than one size category larger than you. If you do not have the Improved Trip feat, or a similar ability, initiating a trip provokes an attack of opportunity from the target of your maneuver.
If your attack exceeds the target's CMD, the target is knocked prone. If your attack fails by 10 or more, you are knocked prone instead. If the target has more than two legs, add +2 to the DC of the combat maneuver attack roll for each additional leg it has. Some creatures—such as oozes, creatures without legs, and flying creatures—cannot be tripped.
You are right that there is no return "trip attack" made on a failed attempt, the character simply falls flat on his face/back or however you wish to describe it.
And I would say that the the first 2 AoO are against full AC, the rest at Prone AC.

Elbedor |

It is a common misconception.
The bold part you're highlighting has to do with the attack roll concerning Combat Maneuvers in general. What happens when the roll exceeds the CMD? The maneuver is a success and has the listed effect.
Under the Trip section it says when the roll exceeds the CMD, then the target is knocked prone. Knocking someone prone via a Trip attack is the definition of successfully tripping someone.
The rules distinguish between a "trip attack" and "to be tripped". If they talk about a "successful trip attempt", they are talking about the roll beating the CMD. If they are talking about "when you're tripped", they are referring to when you're knocked prone.
The Trip feature of weapons and the Trip ability of creatures is just the nutshell.
"If you are tripped during your own trip attempt, you can drop your weapon to avoid being tripped."
Neither of these uses of the word are referring to any combat maneuver being made. There is no roll. "If you are tripped..." and "...avoid being tripped" both mean "If you are knocked prone..." and "...avoid being knocked prone".
There are loads of other examples that show this. I'm not going to post them all here. Anyone is free to search them out themselves. But a few quick examples:
Stay Up: The target uses this ability when it is tripped or otherwise knocked prone. The target steadies itself and stays upright.
During Combat The monk uses Deflect Arrows to close in on ranged combatants. He uses Lunge to keep enemies at bay and flurry of blows to trip multiple opponents if surrounded, allowing him the full benefit of his Combat Reflexes when tripped opponents try to stand. A swift brawler uses Power Attack against enemies he has tripped or stunned, only risking the penalty on his attack rolls when his target's AC has been compromised in some way.
In all of these examples, "trip maneuver" is equated to the roll where "tripped" is equated to being knocked prone (or having your AC compromised in some way)....which is how the rules define these terms pretty consistently.