MathDB

12

Part of 2000 AIME Problems

Problems(2)

Different Values of Function

Source:

1/5/2007
Given a function ff for which f(x)=f(398−x)=f(2158−x)=f(3214−x)f(x)=f(398-x)=f(2158-x)=f(3214-x) holds for all real x,x, what is the largest number of different values that can appear in the list f(0),f(1),f(2),…,f(999)?f(0),f(1),f(2),\ldots,f(999)?
functionalgorithm
Sphere

Source: AIME 2 2000 #12

12/9/2005
The points A,BA, B and CC lie on the surface of a sphere with center OO and radius 20. It is given that AB=13,BC=14,CA=15,AB=13, BC=14, CA=15, and that the distance from OO to triangle ABCABC is mnk,\frac{m\sqrt{n}}k, where m,n,m, n, and kk are positive integers, mm and kk are relatively prime, and nn is not divisible by the square of any prime. Find m+n+k.m+n+k.
geometry3D geometryspherecircumcircletrigonometrytetrahedronnumber theory