Mock AIME I 2015 Problems/Problem 11
=Solution 1
For convenience, let's use instead of
. Define a polynomial
such that
. Let
and
. Then, our polynomial becomes
.
Note that we want to compute
.
From the given information, we know that the coefficient of the term is
, and we also know that
, or in other words,
. By Newton's Sums (since we are given
), we also find that
. Solving this system, we find that
. Thus,
, so our final answer is
.
Solution 2
Let ,
, and
. Then our system becomes
1)
2)
3)
.
2)
Since
, this equation becomes
.
3)
.
Since
, this equation becomes
.
We will now use these equations to solve the problem. Let
, and
. Then we have
.
Solving the system, we find
and
.
Then . So,
.