1997 AIME Problems/Problem 14
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Problem
Let 
and 
be distinct, randomly chosen roots of the equation 
. Let 
be the probability that 
, where 
and 
are relatively prime positive integers. Find 
.
Contents |
Solution
Solution 1
By De Moivre's Theorem, we find that (
)
Now, let be the root corresponding to 
, and let be the root corresponding to 
. The magnitude of 
is therefore:
We need 
. The cosine difference identity simplifies that to 
. Thus, 
.
Therefore, and cannot be more than 
away from each other. This means that for a given value of , there are 
values for that satisfy the inequality; of them 
, and of them 
. Since and must be distinct, can have 
possible values. Therefore, the probability is 
. The answer is then 
.
Solution 2
The solutions of the equation 
are the 
th roots of unity and are equal to 
for 
They are also located at the vertices of a regular -gon that is centered at the origin in the complex plane.
Without loss of generality, let 
Then
We want 
From what we just obtained, this is equivalent to 
This occurs when 
which is satisfied by 
(we don't include 0 because that corresponds to ). So out of the possible 
, work. Thus, 
So our answer is 
See also
| 1997 AIME (Problems • Resources) | ||
| Preceded by Problem 13 | Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
