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Contests
International Contests
IMO Shortlist
2018 IMO Shortlist
A1
Rational FEs
Rational FEs
Source: IMO Shortlist 2018 A1
July 17, 2019
IMO Shortlist
algebra
functional equation
Problem Statement
Let
Q
>
0
\mathbb{Q}_{>0}
Q
>
0
denote the set of all positive rational numbers. Determine all functions
f
:
Q
>
0
→
Q
>
0
f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}
f
:
Q
>
0
→
Q
>
0
satisfying
f
(
x
2
f
(
y
)
2
)
=
f
(
x
)
2
f
(
y
)
f(x^2f(y)^2)=f(x)^2f(y)
f
(
x
2
f
(
y
)
2
)
=
f
(
x
)
2
f
(
y
)
for all
x
,
y
∈
Q
>
0
x,y\in\mathbb{Q}_{>0}
x
,
y
∈
Q
>
0
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