2018 AIME II Problems/Problem 6
Contents
[hide]Problem
A real number is chosen randomly and uniformly from the interval
. The probability that the roots of the polynomial
are all real can be written in the form , where
and
are relatively prime positive integers. Find
.
Solution
The polynomial we are given is rather complicated, so let's use Rational Root Theorem to see if it has any retinal roots. By Rational Root Theorem, are all possible rational roots. Upon plugging these roots into the polynomial,
and
make the polynomial equal 0 and thus, they are roots that we can factor out.
The polynomial becomes:
Since we know and
are real numbers, we only need to focus on the quadratic.
Set the discriminant of the quadratic greater than or equal to 0, to ensure the remaining roots are real.
.
This simplifies to:
or
This means that the interval is the "bad" interval. The length of the interval where
can be chosen from is 38 units long, while the bad interval is 2 units long. Therefore, the "good" interval is 36 units long.
~First
Video Solution
https://www.youtube.com/watch?v=q2oc7n-n6aA ~Shreyas S
See Also:
2018 AIME II (Problems • Answer Key • Resources) | ||
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