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At some point 2008 is written on the board

Source: AIMO 3, German Pre-TST 2009

July 16, 2011

invariantnumber theory unsolvednumber theory

Problem Statement

Initially, on a board there a positive integer. If board contains the number

x,

then we may additionally write the numbers

2x+1

and

\frac{x}{x+2}.

At some point 2008 is written on the board. Prove, that this number was there from the beginning.