Problem:
Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form for some positive integers and . Can you tell me the values of and ?"
After some calculations, Jon says, "There is more than one such polynomial."
Steve says, "You're right. Here is the value of ." He writes down a positive integer and asks, "Can you tell me the value of ?"
Jon says, "There are still two possible values of ."
Find the sum of the two possible values of .
Solution:
Let the roots be , and , with . Then , and , so . The positive integer solutions for are , and . The corresponding values of are and respectively. Because there are two possible values of , it follows that , and the two possible values of are and . The requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions