## 2015 AIME II Problem 6

The other day I came across a problem which reminded me vaguely of the infamous Cheryl’s Birthday Problem (more technical but somewhat easier). Have a go!

2015 AIME II 6. Steve says to Jon, “I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x) = 2x^3 - 2ax^2 + (a^2 - 81)x - c$ for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?

After some calculations, Jon says, “There is more than one such polynomial.”

Steve says, “You’re right. Here is the value of $a$.” He writes down a positive integer and asks, “Can you tell me the value of $c$?”

Jon says, “There are still two possible values of $c$.”

Find the sum of the two possible values of $c$.