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Iran Team Selection Test
2011 Iran Team Selection Test
12
12
Part of
2011 Iran Team Selection Test
Problems
(1)
af(a)+bf(b)+2ab=x^2 for all natural a, b - show that f(a)=a
Source: Iran TST 2011 - Day 4 - Problem 3
5/14/2011
Suppose that
f
:
N
→
N
f : \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
is a function for which the expression
a
f
(
a
)
+
b
f
(
b
)
+
2
a
b
af(a)+bf(b)+2ab
a
f
(
a
)
+
b
f
(
b
)
+
2
ab
for all
a
,
b
∈
N
a,b \in \mathbb{N}
a
,
b
∈
N
is always a perfect square. Prove that
f
(
a
)
=
a
f(a)=a
f
(
a
)
=
a
for all
a
∈
N
a \in \mathbb{N}
a
∈
N
.
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