2003 AIME I Problems/Problem 11
Problem
An angle is chosen at random from the interval
Let
be the probability that the numbers
and
are not the lengths of the sides of a triangle. Given that
where
is the number of degrees in
and
and
are positive integers with
find
Solution
Note that the three expressions are symmetric with respect to interchanging and
, and so the probability is symmetric around
. Thus, take
so that
. Then
is the largest of the three given expressions and those three lengths not forming a triangle is equivalent to a violation of the triangle inequality
This is equivalent to
and, using some of our trigonometric identities, we can re-write this as . Since we've chosen
,
so
The probability that lies in this range is
so that
,
and our answer is
.
See also
2003 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |