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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2005 Moldova Team Selection Test
4
f(n)=n
f(n)=n
Source: Moldova TST 2005
April 10, 2005
function
number theory unsolved
number theory
Problem Statement
Given functions
f
,
g
:
N
∗
→
N
∗
f,g:N^*\rightarrow N^*
f
,
g
:
N
∗
→
N
∗
,
g
g
g
is surjective and
2
f
(
n
)
2
=
n
2
+
g
(
n
)
2
2f(n)^2=n^2+g(n)^2
2
f
(
n
)
2
=
n
2
+
g
(
n
)
2
,
∀
n
>
0
\forall n>0
∀
n
>
0
. Prove that if
∣
f
(
n
)
−
n
∣
≤
2005
n
|f(n)-n|\le2005\sqrt n
∣
f
(
n
)
−
n
∣
≤
2005
n
,
∀
n
>
0
\forall n>0
∀
n
>
0
, then
f
(
n
)
=
n
f(n)=n
f
(
n
)
=
n
for infinitely many
n
n
n
.
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