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.

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