Difference between revisions of "2003 AIME I Problems/Problem 11"
m |
I_like_pie (talk | contribs) |
||
Line 19: | Line 19: | ||
− | The [[probability]] that <math>x</math> lies in this range is <math>\frac 1{45} \cdot \left(\frac 12 \arctan 2\right) = \frac{\arctan 2}{90}</math> so that <math>m = 2</math>, <math>n = 90</math> and our answer is <math> | + | The [[probability]] that <math>x</math> lies in this range is <math>\frac 1{45} \cdot \left(\frac 12 \arctan 2\right) = \frac{\arctan 2}{90}</math> so that <math>m = 2</math>, <math>n = 90</math> and our answer is <math>92</math>. |
== See also == | == See also == |
Revision as of 01:53, 6 November 2006
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
and since we've chosen this means so
or .
The probability that lies in this range is so that , and our answer is .