Difference between revisions of "2003 AIME I Problems/Problem 11"
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== Solution == | == Solution == | ||
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+ | Note that the three expressions are symmetric with respect to interchanging <math>\sin</math> and <math>\cos</math>, and so the probability is symmetric around <math>45^\circ</math>. Thus, take <math>0 < x < 45</math> so that <math>\sin x < \cos x</math>. Then <math>\cos^2 x</math> is the largest of the three given expressions and those three lengths not forming a triangle is equivalent to | ||
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+ | <math>\cos^2 x > \sin^2 x + \sin x \cos x</math> | ||
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+ | This is equivalent to | ||
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+ | <math>\cos^2 x - \sin^2 x > \sin x \cos x</math> | ||
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+ | and, using some of our [[trigonometric identities]] we convert this to | ||
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+ | <math>\cos 2x > \frac 12 \sin 2x</math> and since we've chosen <math>x \in (0, 45)</math> this means | ||
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+ | <math>2 > \tan 2x</math> or <math>x < \frac 12 \arctan 2</math>. | ||
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+ | The probability that <math>x</math> lies in this range is <math>\frac 1{45} \cdot \frac 12 \arctan 2 = \frac{\arctan 2}{90}</math> so that <math>m = 2</math>, <math>n = 90</math> and our answer is <math>092</math>. | ||
== See also == | == See also == | ||
* [[2003 AIME I Problems]] | * [[2003 AIME I Problems]] |
Revision as of 16:31, 11 August 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
This is equivalent to
and, using some of our trigonometric identities we convert this to
and since we've chosen
this means
or
.
The probability that lies in this range is
so that
,
and our answer is
.