Mock AIME I 2012 Problems/Problem 12
Problem
Let be a polynomial of degree 10 satisfying
. Find the maximum possible sum of the coefficients of
.
Solution
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 pair of roots of unity with real parts that differ by
, 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
.