AMC 12A Fall 2021 (Problem 12)
What is the number of terms with rational coefficients among the terms in the expansion of ?
Each term has the form
where .
where .
is rational if and only if
is rational if and only if
Because is even, this is equivalent to
Because is even, this is equivalent to
Therefore,
Hence, let
Hence, let
Since is an integer,
Thus, there are possible values of .
(C)