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Japan MO Finals
2002 Japan MO Finals
4
4
Part of
2002 Japan MO Finals
Problems
(1)
Cyclic inequality
Source: Japan Mathematical Olympiad Finals 2002 , Problem 4
3/22/2006
Let
n
≥
3
n\geq 3
n
≥
3
be natural numbers, and let
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
a_1,\ a_2,\ \cdots,\ a_n,\ \ b_1,\ b_2,\ \cdots,\ b_n
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
be positive numbers such that
a
1
+
a
2
+
⋯
+
a
n
=
1
,
b
1
2
+
b
2
2
+
⋯
+
b
n
2
=
1.
a_1+a_2+\cdots +a_n=1,\ b_1^2+b_2^2+\cdots +b_n^2=1.
a
1
+
a
2
+
⋯
+
a
n
=
1
,
b
1
2
+
b
2
2
+
⋯
+
b
n
2
=
1.
Prove that
a
1
(
b
1
+
a
2
)
+
a
2
(
b
2
+
a
3
)
+
⋯
+
a
n
(
b
n
+
a
1
)
<
1.
a_1(b_1+a_2)+a_2(b_2+a_3)+\cdots +a_n(b_n+a_1)<1.
a
1
(
b
1
+
a
2
)
+
a
2
(
b
2
+
a
3
)
+
⋯
+
a
n
(
b
n
+
a
1
)
<
1.
inequalities
inequalities proposed