1997 AIME Problems/Problem 14
Problem
Let and
be distinct, randomly chosen roots of the equation
. Let
be the probability that
, where
and
are relatively prime positive integers. Find
.
Solution
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 $\displastyle 332$ (Error compiling LaTeX. Unknown error_msg) values for
that satisfy the inequality: $\displastyle 166$ (Error compiling LaTeX. Unknown error_msg) of them are greater than $\displastyle m$ (Error compiling LaTeX. Unknown error_msg), and $\displastyle 166$ (Error compiling LaTeX. Unknown error_msg) are less than $\displastyle m$ (Error compiling LaTeX. Unknown error_msg). Since $\displastyle m$ (Error compiling LaTeX. Unknown error_msg) and $\displastyle n$ (Error compiling LaTeX. Unknown error_msg) must be distinct, $\displastyle n$ (Error compiling LaTeX. Unknown error_msg) can have $\displastyle 1996$ (Error compiling LaTeX. Unknown error_msg) possible values. Therefore, the probability is $\displastyle\frac{332}{1996}=\frac{83}{499}$ (Error compiling LaTeX. Unknown error_msg). The answer is then $\displastyle 499+83=582$ (Error compiling LaTeX. Unknown error_msg)
See also
1997 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |