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National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1999 Dutch Mathematical Olympiad
1
Function with f(mn) = f(m)f(n)
Function with f(mn) = f(m)f(n)
Source: Dutch NMO 1999
October 22, 2005
function
Problem Statement
Let
f
:
Z
→
{
−
1
,
1
}
f: \mathbb{Z} \rightarrow \{-1,1\}
f
:
Z
→
{
−
1
,
1
}
be a function such that
f
(
m
n
)
=
f
(
m
)
f
(
n
)
,
∀
m
,
n
∈
Z
.
f(mn) =f(m)f(n),\ \forall m,n \in \mathbb{Z}.
f
(
mn
)
=
f
(
m
)
f
(
n
)
,
∀
m
,
n
∈
Z
.
Show that there exists a positive integer
a
a
a
such that
1
≤
a
≤
12
1 \leq a \leq 12
1
≤
a
≤
12
and
f
(
a
)
=
f
(
a
+
1
)
=
1
f(a) = f(a + 1) = 1
f
(
a
)
=
f
(
a
+
1
)
=
1
.
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