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Calculator with 2 operations, possible to make r

Source: Lusophon Mathematical Olympiad 2023 Problem 6

May 27, 2023

algebra

Problem Statement

A calculator has two operations

\dfrac{x}{x+1}

and

B

and initially shows the number

1

. Operation

A

turns

x

into

x+1

and operation B turns

x

into

\dfrac{x}{x+1}

. a) Show all the ways we can get the number

\dfrac{20}{23}

. b) For every rational

r \neq 1

, determine if it is possible to get

r

using only operations

A

and

B

.

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