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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2021 Vietnam National Olympiad
1
VMO 2021, P1
VMO 2021, P1
Source:
December 25, 2020
algebra
Problem Statement
Let
(
x
n
)
(x_n)
(
x
n
)
define by
x
1
∈
(
0
;
1
2
)
x_1\in \left(0;\dfrac{1}{2}\right)
x
1
∈
(
0
;
2
1
)
and
x
n
+
1
=
3
x
n
2
−
2
n
x
n
3
x_{n+1}=3x_n^2-2nx_n^3
x
n
+
1
=
3
x
n
2
−
2
n
x
n
3
for all
n
≥
1
n\ge 1
n
≥
1
. a) Prove that
(
x
n
)
(x_n)
(
x
n
)
convergence to
0
0
0
.b) For each
n
≥
1
n\ge 1
n
≥
1
, let
y
n
=
x
1
+
2
x
2
+
⋯
+
n
x
n
y_n=x_1+2x_2+\cdots+n x_n
y
n
=
x
1
+
2
x
2
+
⋯
+
n
x
n
. Prove that
(
y
n
)
(y_n)
(
y
n
)
has a limit.
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