Mock AIME I 2012 Problems/Problem 12
Let be a polynomial of degree 10 satisfying . Find the maximum possible sum of the coefficients of .
Notice that if is a root of , then must be a root of and must be a root of . But then continuing this, and must be roots of for all . Since a polynomial has finitely many roots, and must be roots of unity so that the above two sets contain finitely many elements. But there is a unique such pair of roots of unity, making . Then the disjoint union of the two sets above is , the minimal polynomial for which is . Since any power of this base polynomial will work, , making the sum of coefficients .