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National and Regional Contests
Turkey Contests
Turkey Team Selection Test
1989 Turkey Team Selection Test
1
1
Part of
1989 Turkey Team Selection Test
Problems
(1)
f(m, m+k) = f(m,k)
Source: Turkey TST 1989 - P1
9/11/2013
Let
Z
+
\mathbb{Z}^+
Z
+
denote the set of positive integers. Find all functions
f
:
Z
+
×
Z
+
→
Z
+
f: \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+
f
:
Z
+
×
Z
+
→
Z
+
such that [*]
f
(
m
,
m
)
=
m
f(m,m)=m
f
(
m
,
m
)
=
m
[*]
f
(
m
,
k
)
=
f
(
k
,
m
)
f(m,k) = f(k,m)
f
(
m
,
k
)
=
f
(
k
,
m
)
[*]
f
(
m
,
m
+
k
)
=
f
(
m
,
k
)
f(m, m+k) = f(m,k)
f
(
m
,
m
+
k
)
=
f
(
m
,
k
)
, for each
m
,
k
∈
Z
+
m,k \in \mathbb{Z}^+
m
,
k
∈
Z
+
.
function
algorithm
number theory proposed
number theory