2013 AIME II Problems/Problem 12
Let be the set of all polynomials of the form , where , , and are integers. Find the number of polynomials in such that each of its roots satisfies either or .
Solution
Every cubic in real coefficients has to have either three real roots or one real and two complex roots which are conjugates. This follows from Viète's identities. Factorise the polynomial , where omega* is the complex conjugate of omega. r is the real root which must be -20, 20, -13, or 13, and it doesn't matter which. 20 or 13. Let . Viète tells us that ), but (i.e., adding the conjugates cancels the imaginary part). a the quadratic coefficient must be an integer so is some integer over 2. 20 or 13 so you have a bound on : either or . Don't forget zero! You have the magnitude so determines totally (you can solve for the imaginary part) and determines . Now just count: 4 possibilities for the real root times [(26+1) possibilities if + (40+1) possibilities if = 272.