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Ecuador Mathematical Olympiad (OMEC)
2023 Ecuador NMO (OMEC)
3
3
Part of
2023 Ecuador NMO (OMEC)
Problems
(1)
A 'polynomial tree', with 'a' as its leaves
Source: OMEC Ecuador National Olympiad Final Round 2023 N3 P3 day 1
11/5/2024
We define a sequence of numbers
a
n
a_n
a
n
such that
a
0
=
1
a_0=1
a
0
=
1
and for all
n
≥
0
n\ge0
n
≥
0
:
2
a
n
+
1
3
+
2
a
n
3
=
3
a
n
+
1
2
a
n
+
3
a
n
+
1
a
n
2
2a_{n+1} ^3 + 2a_n ^3 = 3 a_{n +1} ^2 a_n + 3a_{n+1}a_n^2
2
a
n
+
1
3
+
2
a
n
3
=
3
a
n
+
1
2
a
n
+
3
a
n
+
1
a
n
2
Find the sum of all
a
2023
a_{2023}
a
2023
's possible values.
algebra
polynomial