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2019 CHMMC (Fall)
8
2019 Fall Team #8
2019 Fall Team #8
Source:
April 17, 2022
algebra
Problem Statement
Consider an infinite sequence of reals
x
1
,
x
2
,
x
3
,
.
.
.
x_1, x_2, x_3, ...
x
1
,
x
2
,
x
3
,
...
such that
x
1
=
1
x_1 = 1
x
1
=
1
,
x
2
=
2
3
3
x_2 =\frac{2\sqrt3}{3}
x
2
=
3
2
3
and with the recursive relationship
n
2
(
x
n
−
x
n
−
1
−
x
n
−
2
)
−
n
(
3
x
n
+
2
x
n
−
1
+
x
n
−
2
)
+
(
x
n
x
n
−
1
x
n
−
2
+
2
x
n
)
=
0.
n^2 (x_n - x_{n-1} - x_{n-2}) - n(3x_n + 2x_{n-1} + x_{n-2}) + (x_nx_{n-1}x_{n-2} + 2x_n) = 0.
n
2
(
x
n
−
x
n
−
1
−
x
n
−
2
)
−
n
(
3
x
n
+
2
x
n
−
1
+
x
n
−
2
)
+
(
x
n
x
n
−
1
x
n
−
2
+
2
x
n
)
=
0.
Find
x
2019
x_{2019}
x
2019
.
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