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Putnam
2003 Putnam
2
Putnam 2003 A2
Putnam 2003 A2
Source:
June 22, 2011
Putnam
inequalities
college contests
Putnam inequalities
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
n
a_1, a_2, \cdots , a_n
a
1
,
a
2
,
⋯
,
a
n
and
b
1
,
b
2
,
⋯
,
b
n
b_1, b_2,\cdots, b_n
b
1
,
b
2
,
⋯
,
b
n
be nonnegative real numbers. Show that
(
a
1
a
2
⋯
a
n
)
1
/
n
+
(
b
1
b
2
⋯
b
n
)
1
/
n
≤
(
(
a
1
+
b
1
)
(
a
2
+
b
2
)
⋯
(
a
n
+
b
n
)
)
1
/
n
(a_1a_2 \cdots a_n)^{1/n}+ (b_1b_2 \cdots b_n)^{1/n} \le ((a_1 + b_1)(a_2 + b_2) \cdots (a_n + b_n))^{1/n}
(
a
1
a
2
⋯
a
n
)
1/
n
+
(
b
1
b
2
⋯
b
n
)
1/
n
≤
((
a
1
+
b
1
)
(
a
2
+
b
2
)
⋯
(
a
n
+
b
n
)
)
1/
n
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