Problem:
An angle x is chosen at random from the interval 0∘<x<90∘. Let p be the probability that the numbers sin2x,cos2x, and sinxcosx are not the lengths of the sides of a triangle. Given that p=d/n, where d is the number of degrees in arctanm and m and n are positive integers with m+n<1000, find m+n.
Solution:
Because cos(90∘−x)=sinx and sin(90∘−x)=cosx, it suffices to consider x in the interval 0∘<x≤45∘. For such x,
cos2x≥sinxcosx≥sin2x
so the three numbers are not the lengths of the sides of a triangle if and only if
cos2x≥sin2x+sinxcosx
which is equivalent to cos2x≥21sin2x, or tan2x≤2. Because the tangent function is increasing in the interval 0∘≤x≤45∘, this inequality is equivalent to x≤21(arctan2)∘. It follows that
p=45∘21(arctan2)∘=90∘(arctan2)∘
so m+n=92.
The problems on this page are the property of the MAA's American Mathematics Competitions