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.

Advertisements
This entry was posted in Grade 12, USA and tagged , , , . Bookmark the permalink.

Leave a Reply

Please log in using one of these methods to post your comment:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s