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Canada National Olympiad
2021 Canada National Olympiad
4
4
Part of
2021 Canada National Olympiad
Problems
(1)
Canadian MO 2021 P4
Source:
3/12/2021
A function
f
f
f
from the positive integers to the positive integers is called Canadian if it satisfies
gcd
(
f
(
f
(
x
)
)
,
f
(
x
+
y
)
)
=
gcd
(
x
,
y
)
\gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y)
g
cd
(
f
(
f
(
x
))
,
f
(
x
+
y
)
)
=
g
cd
(
x
,
y
)
for all pairs of positive integers
x
x
x
and
y
y
y
.Find all positive integers
m
m
m
such that
f
(
m
)
=
m
f(m)=m
f
(
m
)
=
m
for all Canadian functions
f
f
f
.
number theory
algebra