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f(x_1) = x_2, f(x_2) = x_3 , f(x_3) = x_1 if f(x) = 1/(ax+b)

Source: 1993 Swedish Mathematical Competition p6

April 2, 2021

functionalgebra

Problem Statement

For real numbers

a

and

b

define

f(x) = \frac{1}{ax+b}

. For which

a

and

b

are there three distinct real numbers

x_1,x_2,x_3

such that

f(x_1) = x_2

f(x_2) = x_3

and

f(x_3) = x_1