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4
2002 BAMO p4 inequality with largest odd divisor
2002 BAMO p4 inequality with largest odd divisor
Source:
August 26, 2019
inequalities
number theory
divisor
Problem Statement
For
n
≥
1
n \ge 1
n
≥
1
, let
a
n
a_n
a
n
be the largest odd divisor of
n
n
n
, and let
b
n
=
a
1
+
a
2
+
.
.
.
+
a
n
b_n = a_1+a_2+...+a_n
b
n
=
a
1
+
a
2
+
...
+
a
n
. Prove that
b
n
≥
n
2
+
2
3
b_n \ge \frac{ n^2 + 2}{3}
b
n
≥
3
n
2
+
2
, and determine for which
n
n
n
equality holds. For example,
a
1
=
1
,
a
2
=
1
,
a
3
=
3
,
a
4
=
1
,
a
5
=
5
,
a
6
=
3
a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3
a
1
=
1
,
a
2
=
1
,
a
3
=
3
,
a
4
=
1
,
a
5
=
5
,
a
6
=
3
, thus
b
6
=
1
+
1
+
3
+
1
+
5
+
3
=
14
≥
6
2
+
2
3
=
12
2
3
b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23
b
6
=
1
+
1
+
3
+
1
+
5
+
3
=
14
≥
3
6
2
+
2
=
12
3
2
.
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