Mock AIME II 2012 Problems/Problem 12
Problem
Let . Assume the value of has three real solutions . If , where and are relatively prime positive integers, find .
Solution
Let . Then and . From this, we have the system
Substituting the first equation into the second, we obtain
Plugging this into the third equation yields .
Thus, . Note that our three real roots multiply to . However, since , we need to multiply by , so our is
We need . Using vieta’s and making sure we count for each factor of we divided off, we have .
Our answer is , thus .
Solution 2
Let and , where . Then, it is obvious that .
We first focus on the first equality: . This may be simplified using our logarithmic properties:
Now, let's focus on the last expression: note that,
We can equate all of these expressions:
Multiplying all expressions by gives us
Now, from our first equality we obtain
Since , we may safely divide by :
From the first and last expressions we have:
Equating our expressions for gives
Since , we may safely divide by :
By Vieta's formulas, we must have and . Dividing the former by the latter gives
and hence .
~FIREDRAGONMATH16