There are other alternate valuation methods out there that may or may not increase your cost basis and help you. I'm not all that privy to the inner workings of these other methods, as they can get rather complicated and require a complex paper trail to substantiate if audited on it. Therefore, I'm keeping it simple.
Typically with inherited property, the cost basis is the FMV of the property on the date the previous owner passed. If audited on that valuation, some kind of documentation may need to be provided in order to support said valuation.
If each beneficiary gets 1/3 of the property then the cost basis for each of them is 1/3 of the previously established FMV.
If one beneficiary gifts their share to another beneficiary then that is a gift. If the FMV of their share is more than $17,000 then the giver (not the recipient) is required to file IRS Form 709-Gift Tax Return with the IRS. If the value of the gift is less than $11.5M (and I'm sure it is) then no taxes will be paid. But like I stated already, if the value is more than $17K the IRS reporting on form 709 by the giver is still required.
Say the house is worth $450k. Instead of me buying him out for 1/3 x 450 = 150, he would only charge me 100.
Using your numbers, they would still be gifting you $50K. Since that's more than $17K they would be required to file the form 709 with the IRS for the $50K they are gifting.
Assuming the most simplistic valuation method is used, your cost basis would be (using your numbers):
$150K for the portion you inherited plus
$150K for the portion gifted to you plus
$100K for the portion you purchased plus
$50K for the portion gifted to you.
Total would be $450K
