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balkan sequence
Source: Balkan Math Olympiad BkMO 2004, problem 1
5/7/2004
The sequence
{
a
n
}
n
≥
0
\{a_n\}_{n\geq 0}
{
a
n
}
n
≥
0
of real numbers satisfies the relation:
a
m
+
n
+
a
m
−
n
−
m
+
n
−
1
=
1
2
(
a
2
m
+
a
2
n
)
a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n})
a
m
+
n
+
a
m
−
n
−
m
+
n
−
1
=
2
1
(
a
2
m
+
a
2
n
)
for all non-negative integers
m
m
m
and
n
n
n
,
m
≥
n
m \ge n
m
≥
n
. If
a
1
=
3
a_1 = 3
a
1
=
3
find
a
2004
a_{2004}
a
2004
.
induction
algebra proposed
algebra