MathDB

11

Part of 2003 AIME Problems

Problems(2)

Sides of a triangle

Source:

5/12/2006
An angle xx is chosen at random from the interval 0∘<x<90∘0^\circ < x < 90^\circ. Let pp be the probability that the numbers sin⁡2x\sin^2 x, cos⁡2x\cos^2 x, and sin⁡xcos⁡x\sin x \cos x are not the lengths of the sides of a triangle. Given that p=d/np = d/n, where dd is the number of degrees in arctan⁡m\arctan m and mm and nn are positive integers with m+n<1000m + n < 1000, find m+nm + n.
probabilitytrigonometrysymmetryinequalitiesalgebrafunctiondomain
Area of a Subtriangle

Source:

12/26/2006
Triangle ABCABC is a right triangle with AC=7,AC=7, BC=24,BC=24, and right angle at C.C. Point MM is the midpoint of AB,AB, and DD is on the same side of line ABAB as CC so that AD=BD=15.AD=BD=15. Given that the area of triangle CDMCDM may be expressed as mnp,\frac{m\sqrt{n}}{p}, where m,m, n,n, and pp are positive integers, mm and pp are relatively prime, and nn is not divisible by the square of any prime, find m+n+p.m+n+p.
geometrycircumcircletrigonometrynumber theoryrelatively primetrig identitiesLaw of Cosines