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Lusophon Mathematical Olympiad
2023 Lusophon Mathematical Olympiad
6
6
Part of
2023 Lusophon Mathematical Olympiad
Problems
(1)
Calculator with 2 operations, possible to make r
Source: Lusophon Mathematical Olympiad 2023 Problem 6
5/27/2023
A calculator has two operations
A
A
A
and
B
B
B
and initially shows the number
1
1
1
. Operation
A
A
A
turns
x
x
x
into
x
+
1
x+1
x
+
1
and operation B turns
x
x
x
into
x
x
+
1
\dfrac{x}{x+1}
x
+
1
x
. a) Show all the ways we can get the number
20
23
\dfrac{20}{23}
23
20
. b) For every rational
r
≠
1
r \neq 1
r
=
1
, determine if it is possible to get
r
r
r
using only operations
A
A
A
and
B
B
B
.
algebra