2003 AIME I Problems/Problem 11

Problem

An angle $x$ is chosen at random from the interval $0^\circ < x < 90^\circ.$ Let $p$ be the probability that the numbers $\sin^2 x, \cos^2 x,$ and $\sin x \cos x$ are not the lengths of the sides of a triangle. Given that $p = d/n,$ where $d$ is the number of degrees in $\arctan m$ and $m$ and $n$ are positive integers with $m + n < 1000,$ find $m + n.$

Solution

Note that the three expressions are symmetric with respect to interchanging $\sin$ and $\cos$ , and so the probability is symmetric around $45^\circ$ . Thus, take $0 < x < 45$ so that $\sin x < \cos x$ . Then $\cos^2 x$ 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

$\cos^2 x > \sin^2 x + \sin x \cos x$

This is equivalent to

$\cos^2 x - \sin^2 x > \sin x \cos x$

and, using some of our trigonometric identities, we can re-write this as

$\cos 2x > \frac 12 \sin 2x$ and since we've chosen $x \in (0, 45)$ this means $\cos 2x > 0$ so

$2 > \tan 2x$ or $x < \frac 12 \arctan 2$ .

The probability that $x$ lies in this range is $\frac 1{45} \cdot \left(\frac 12 \arctan 2\right) = \frac{\arctan 2}{90}$ so that $m = 2$ , $n = 90$ and our answer is $092$ .

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2003 AIME I Problems/Problem 11

Problem

Solution

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