MathDB

14

Part of 2002 AIME Problems

Problems(2)

Sets

Source: AIME 2002 #14

11/28/2005
A set S\mathcal{S} of distinct positive integers has the following property: for every integer xx in S,\mathcal{S}, the arithmetic mean of the set of values obtained by deleting xx from S\mathcal{S} is an integer. Given that 1 belongs to S\mathcal{S} and that 2002 is the largest element of S,\mathcal{S}, what is the greatet number of elements that S\mathcal{S} can have?
AIMEAIME IAIME IIUSASetsmodular arithmetic
Right Triangle and Circle

Source:

12/28/2006
The perimeter of triangle APMAPM is 152,152, and the angle PAMPAM is a right angle. A circle of radius 1919 with center OO on AP‾\overline{AP} is drawn so that it is tangent to AM‾\overline{AM} and PM‾.\overline{PM}. Given that OP=m/n,OP=m/n, where mm and nn are relatively prime positive integers, find m+n.m+n.
geometryperimetergeometric transformationdilationtrigonometrynumber theoryrelatively prime