Mock AIME 2 Pre 2005 Problems/Problem 9
Solution
We begin by determining the value of . Experimenting, we find the first few
s:
We observe that because ,
will be determined by the base 2 expansion of i. Specifically, every 1 in the
s digit of the expansion corresponds to adding
to
. Since
base 2,
Now we look for ways to attain an element with degree . Since each sum of powers of 3 is unique, there is only one; namely, take the x element for every binomial with a degree of one of the added powers of 3 in
, and the 1 for all else. Finally, since the coefficients of the x elements are equal to the degree to which the 3 is raised, we conclude