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 $$m + n.$$

(第二十一届AIME1 2003 第11题)