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Rioplatense Mathematical Olympiad, Level 3
1992 Rioplatense Mathematical Olympiad, Level 3
1
1
Part of
1992 Rioplatense Mathematical Olympiad, Level 3
Problems
(1)
\log_{f(1)}f(x) is perfect square if f(x + y)f(x-y) = (f(x)f(y))^2
Source: II - Rioplatense 1992 L3 P1
9/19/2022
Let
f
:
Z
→
N
−
{
0
}
f:Z \to N -\{0\}
f
:
Z
→
N
−
{
0
}
such that:
f
(
x
+
y
)
f
(
x
−
y
)
=
(
f
(
x
)
f
(
y
)
)
2
f(x + y)f(x-y) = (f(x)f(y))^2
f
(
x
+
y
)
f
(
x
−
y
)
=
(
f
(
x
)
f
(
y
)
)
2
and
f
(
1
)
≠
1
f(1)\ne 1
f
(
1
)
=
1
.Provethat
log
f
(
1
)
f
(
z
)
\log_{f(1)}f(z)
lo
g
f
(
1
)
f
(
z
)
is a perfect square for every integer
z
z
z
.
number theory
functional
functional equation