MathDB

10

Part of 2000 AIME Problems

Problems(2)

Sequence of Numbers

Source:

1/5/2007
A sequence of numbers x1,x2,x3,…,x100x_{1},x_{2},x_{3},\ldots,x_{100} has the property that, for every integer kk between 11 and 100,100, inclusive, the number xkx_{k} is kk less than the sum of the other 9999 numbers. Given that x50=m/n,x_{50}=m/n, where mm and nn are relatively prime positive integers, find m+n.m+n.
number theoryrelatively primeAIMEAIME 2000
Inscribed Circle

Source:

12/9/2005
A circle is inscribed in quadrilateral ABCD,ABCD, tangent to AB‾\overline{AB} at PP and to CD‾\overline{CD} at Q.Q. Given that AP=19,PB=26,CQ=37,AP=19, PB=26, CQ=37, and QD=23,QD=23, find the square of the radius of the circle.
trigonometrygeometrytrapezoidangle bisector