8

Part of 1988 Irish Math Olympiad

Problems(1)

Sequence with all Integers

Source: 1988 IrMO Paper 1 Problem 8

x_1,x_2,x_3,\dots

be sequence of nonzero real numbers satisfying x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}, n=3,4,5,\dots Establish necessary and sufficient conditions on

x_1,x_2

x_n

to be an integer for infinitely many values of

n