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Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1992 Vietnam National Olympiad
3
tend to infty
tend to infty
Source: 30-th Vietnamese Mathematical Olympiad 1992
February 17, 2007
limit
inequalities
calculus
calculus computations
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals and sequences
{
a
n
}
,
{
b
n
}
,
{
c
n
}
\{a_{n}\},\{b_{n}\},\{c_{n}\}
{
a
n
}
,
{
b
n
}
,
{
c
n
}
defined by
a
k
+
1
=
a
k
+
2
b
k
+
c
k
,
b
k
+
1
=
b
k
+
2
c
k
+
a
k
,
c
k
+
1
=
c
k
+
2
a
k
+
b
k
a_{k+1}=a_{k}+\frac{2}{b_{k}+c_{k}},b_{k+1}=b_{k}+\frac{2}{c_{k}+a_{k}},c_{k+1}=c_{k}+\frac{2}{a_{k}+b_{k}}
a
k
+
1
=
a
k
+
b
k
+
c
k
2
,
b
k
+
1
=
b
k
+
c
k
+
a
k
2
,
c
k
+
1
=
c
k
+
a
k
+
b
k
2
for all
k
=
0
,
1
,
2
,
.
.
.
k=0,1,2,...
k
=
0
,
1
,
2
,
...
. Prove that
lim
k
→
+
∞
a
k
=
lim
k
→
+
∞
b
k
=
lim
k
→
+
∞
c
k
=
+
∞
\lim_{k\to+\infty}a_{k}=\lim_{k\to+\infty}b_{k}=\lim_{k\to+\infty}c_{k}=+\infty
lim
k
→
+
∞
a
k
=
lim
k
→
+
∞
b
k
=
lim
k
→
+
∞
c
k
=
+
∞
.
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