MathDB

3

Part of 1995 Cono Sur Olympiad

Problems(2)

Circles inside a rectangle

Source: Cono Sur Olympiad, 1995, Problem 3

5/28/2006
Let ABCDABCD be a rectangle with: AB=aAB=a, BC=bBC=b. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides ABAB and ADAD,the other is tangent to the sides CBCB and CDCD. 1. Find the distance between the centers of the circles(using aa and bb). 2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.
geometryrectanglequadraticsalgebracono sur
F(n)

Source: Cono Sur Olympiad, 1995, Problem #6

5/28/2006
Let nn be a natural number and f(n)=2n−1995⌊n1000⌋f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor(⌊\lfloor ⌋\rfloor denotes the floor function). 1. Show that if for some integer rr: f(f(f...f(n)...))=1995f(f(f...f(n)...))=1995 (where the function ff is applied rr times), then nn is multiple of 19951995. 2. Show that if nn is multiple of 1995, then there exists r such that:f(f(f...f(n)...))=1995f(f(f...f(n)...))=1995 (where the function ff is applied rr times). Determine rr if n=1995.500=997500n=1995.500=997500
floor functionfunctionmodular arithmeticinequalitiesnumber theoryrelatively prime