MathDB

9

Part of 2001 AIME Problems

Problems(2)

Area Ratio

Source:

12/6/2005
In triangle ABCABC, AB=13,AB=13, BC=15BC=15 and CA=17.CA=17. Point DD is on AB‾,\overline{AB}, EE is on BC‾,\overline{BC}, and FF is on CA‾.\overline{CA}. Let AD=p⋅AB,AD=p\cdot AB, BE=q⋅BC,BE=q\cdot BC, and CF=r⋅CA,CF=r\cdot CA, where p,p, q,q, and rr are positive and satisfy p+q+r=2/3p+q+r=2/3 and p2+q2+r2=2/5.p^2+q^2+r^2=2/5. The ratio of the area of triangle DEFDEF to the area of triangle ABCABC can be written in the form m/n,m/n, where mm and nn are relatively prime positive integers. Find m+n.m+n.
geometryratioanalytic geometrytrigonometrynumber theoryrelatively prime
Coloring a square grid

Source:

12/28/2006
Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
probabilitygeometryrectanglenumber theoryrelatively prime