A good path?

Oregon State is undefeated against UCLA to open a Pac-10/12 Tournament.  (Having said that, Oregon State has never played UCLA in the First Round of the Pac-10/12 Tournament, only playing UCLA in the Quarterfinals after getting a bye) Oregon State is also undefeated against Oregon outside of the First Round.

Looking to the Pac-12 Semifinal, Oregon State is 0-6 all-time against Arizona.  Every time that Oregon State lost in the Semifinal, the winner was Arizona.  Otherwise, Oregon State beat Wazzu in overtime in 1988 and beat Oregon in 2021.

A fun fact is that the only team to defeat the #1 Pac-10/12 team in the Quarterfinals/Semifinals in the past 16 seasons?  Oregon State.  Oregon State beat #1 Washington in 2012 and #1 Oregon in 2021.  In every other season, the #1 Pac-12 team played in the Championship Game (except, of course, in 2020, after every game after the First Round was wiped out by COVID-19).

Having said that, in four of the six losses to Arizona, Arizona was the #1 overall seed.  #1 Arizona most recently defeated Oregon State in 2011.  The Beavs are undefeated against the #1 Pac-12 team since the loss to Arizona in 2011.

Oregon State is 3-1 all-time against Washington and undefeated after the First Round.  USC beat Oregon State in the 2018 Pac-12 Quarterfinals in the two teams' only Pac-12 Tournament meeting.  USC played #1 Arizona in the Pac-12 Championship Game in 2018 and lost.

On the other side of the bracket, Oregon State beat Colorado in the Pac-12 Championship Game in 2021.  Oregon State has only played in one other Pac-10 Championship Game, losing to #1 Arizona.  

The only team on the other side of the bracket that Oregon State has played and beat after the Quarterfinals in the Pac-10/12 Tournament all-time is Wazzu.  The Beavs are undefeated against the Cougs all-time in the Pac-10/12 Tournament.  Oregon State beat Cal in the 1989 Quarterfinals but lost every other meeting: 1987, 2003, 2007, 2016, and 2017.  The higher-seeded team has always won the Cal-Oregon State game.  Oregon State is undefeated against Utah in the Pac-12 Tournament. 

The only other teams that have not been mentioned so far are Stanford and Arizona State.  Oregon State is 1-1 against Stanford and 2-2 against Arizona State.  Notably, the lower-rated team has upset the higher-rated team in each Oregon State-Stanford meeting.  Oregon State infamously lost the 1990 Pac-10 Quarterfinals against Arizona State, upset Arizona State in 2006, beat Arizona State in 2016, and lost to Arizona State last year.

For money reasons, hopefully, a team that finished fourth or lower wins the Pac-12.