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Problems
Contests
International Contests
Baltic Way
2016 Baltic Way
10
Sequence and inequality
Sequence and inequality
Source: Baltic Way 2016, Problem 10
November 5, 2016
algebra
Problem Statement
Let
a
0
,
1
,
a
0
,
2
,
.
.
.
,
a
0
,
2016
a_{0,1}, a_{0,2}, . . . , a_{0, 2016}
a
0
,
1
,
a
0
,
2
,
...
,
a
0
,
2016
be positive real numbers. For
n
≥
0
n\geq 0
n
≥
0
and
1
≤
k
<
2016
1 \leq k < 2016
1
≤
k
<
2016
set
a
n
+
1
,
k
=
a
n
,
k
+
1
2
a
n
,
k
+
1
and
a
n
+
1
,
2016
=
a
n
,
2016
+
1
2
a
n
,
1
.
a_{n+1,k} = a_{n,k} +\frac{1}{2a_{n,k+1}} \ \ \text{and} \ \ a_{n+1,2016} = a_{n,2016} +\frac{1}{2a_{n,1}}.
a
n
+
1
,
k
=
a
n
,
k
+
2
a
n
,
k
+
1
1
and
a
n
+
1
,
2016
=
a
n
,
2016
+
2
a
n
,
1
1
.
Show that
max
1
≤
k
≤
2016
a
2016
,
k
>
44.
\max_{1\leq k \leq 2016} a_{2016,k} > 44.
max
1
≤
k
≤
2016
a
2016
,
k
>
44.
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