MathDB

2

Part of 2012 CentroAmerican

Problems(2)

Least amount of money

Source: Centroamerican 2012, Problem 5

6/20/2012
Alexander and Louise are a pair of burglars. Every morning, Louise steals one third of Alexander's money, but feels remorse later in the afternoon and gives him half of all the money she has. If Louise has no money at the beginning and starts stealing on the first day, what is the least positive integer amount of money Alexander must have so that at the end of the 2012th day they both have an integer amount of money?
inductionnumber theory unsolvednumber theory
Line passes through midpoint

Source: Centroamerican 2012, Problem 2

6/19/2012
Let γ\gamma be the circumcircle of the acute triangle ABCABC. Let PP be the midpoint of the minor arc BCBC. The parallel to ABAB through PP cuts BC,ACBC, AC and γ\gamma at points R,SR,S and TT, respectively. Let K≡AP∩BTK \equiv AP \cap BT and L≡BS∩ARL \equiv BS \cap AR. Show that KLKL passes through the midpoint of ABAB if and only if CS=PRCS = PR.
geometrycircumcircletrapezoidperpendicular bisectorgeometry unsolved