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National and Regional Contests
Romania Contests
District Olympiad
2003 District Olympiad
2
Famous recurrence
Famous recurrence
Source: RMO 2003, District Round
May 29, 2006
function
algebra proposed
algebra
Problem Statement
Find all functions
f
:
N
∗
→
M
\displaystyle f : \mathbb N^\ast \to M
f
:
N
∗
→
M
such that
1
+
f
(
n
)
f
(
n
+
1
)
=
2
n
2
(
f
(
n
+
1
)
−
f
(
n
)
)
,
∀
n
∈
N
∗
,
\displaystyle 1 + f(n) f(n+1) = 2 n^2 \left( f(n+1) - f(n) \right), \, \forall n \in \mathbb N^\ast ,
1
+
f
(
n
)
f
(
n
+
1
)
=
2
n
2
(
f
(
n
+
1
)
−
f
(
n
)
)
,
∀
n
∈
N
∗
,
in each of the following situations: (a)
M
=
N
\displaystyle M = \mathbb N
M
=
N
; (b)
M
=
Q
\displaystyle M = \mathbb Q
M
=
Q
. Dinu Şerbănescu
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