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Odd positive integer functions

Source: USAMO 1993

October 27, 2005
functioninductionnumber theoryrelatively primealgebra unsolvedalgebra

Problem Statement

Let  a,b \, a,b \, be odd positive integers. Define the sequence  (fn) \, (f_n ) \, by putting  f1=a,\, f_1 = a, f2=b, f_2 = b, \, and by letting  fn \, f_n \, for  n≥3 \, n \geq 3 \, be the greatest odd divisor of  fn−1+fn−2\, f_{n-1} + f_{n-2}. Show that  fn \, f_n \, is constant for  n \, n \, sufficiently large and determine the eventual value as a function of  a \, a \, and  b\, b.
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