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 | ||
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