MathDB

12

Part of 2004 AIME Problems

Problems(2)

Log area.

Source:

11/27/2005
Let SS be the set of ordered pairs (x,y)(x, y) such that 0<x≤10<x\le 1, 0<y≤10<y\le 1, and [log⁡2(1x)]\left[\log_2{\left(\frac 1x\right)}\right] and [log⁡5(1y)]\left[\log_5{\left(\frac 1y\right)}\right] are both even. Given that the area of the graph of SS is m/nm/n, where mm and nn are relatively prime positive integers, find m+nm+n. The notation [z][z] denotes the greatest integer that is less than or equal to zz.
logarithmsgeometryfloor functioninequalitiesrectanglenumber theoryrelatively prime
Isosceles Trapezoid

Source:

11/17/2005
Let ABCDABCD be an isosceles trapezoid, whose dimensions are AB=6AB = 6, BC=5=DABC=5=DA, and CD=4CD=4. Draw circles of radius 3 centered at AA and BB, and circles of radius 2 centered at CC and DD. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is −k+mnp\frac{-k+m\sqrt{n}}p, where kk, mm, nn, and pp are positive integers, nn is not divisible by the square of any prime, and kk and pp are relatively prime. Find k+m+n+pk+m+n+p.
geometrytrapezoidanalytic geometryquadraticsUSAMTSalgebra