MathDB

12

Part of 2006 AIME Problems

Problems(2)

Trigonometry

Source: 2006 AIME A Problem 12

3/9/2006
Find the sum of the values of xx such that cos⁡33x+cos⁡35x=8cos⁡34xcos⁡3x\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x, where xx is measured in degrees and 100<x<200100< x< 200.
trigonometryfunctionAMC
Geometry yet again

Source: 2006 AIME II 12

3/28/2006
Equilateral △ABC\triangle ABC is inscribed in a circle of radius 2. Extend AB‾\overline{AB} through BB to point DD so that AD=13AD=13, and extend AC‾\overline{AC} through CC to point EE so that AE=11AE=11. Through DD, draw a line l1l_1 parallel to AE‾\overline{AE}, and through EE, draw a line l2{l}_2 parallel to AD‾\overline{AD}. Let FF be the intersection of l1{l}_1 and l2{l}_2. Let GG be the point on the circle that is collinear with AA and FF and distinct from AA. Given that the area of △CBG\triangle CBG can be expressed in the form pqr\frac{p\sqrt{q}}{r}, where pp, qq, and rr are positive integers, pp and rr are relatively prime, and qq is not divisible by the square of any prime, find p+q+rp+q+r.
geometrytrigonometryAMCAIMEparallelogramgeometric transformationhomothety