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National and Regional Contests
China Contests
(China) National High School Mathematics League
2018 China Second Round Olympiad
1
China Second Round Olympiad 2018Test 2 Q1
China Second Round Olympiad 2018Test 2 Q1
Source:
September 9, 2018
inequalities
China
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
,
A
,
B
a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n,A,B
a
1
,
a
2
,
⋯
,
a
n
,
b
1
,
b
2
,
⋯
,
b
n
,
A
,
B
are positive reals such that
a
i
≤
b
i
,
a
i
≤
A
a_i\leq b_i,a_i\leq A
a
i
≤
b
i
,
a
i
≤
A
(
i
=
1
,
2
,
⋯
,
n
)
(i=1,2,\cdots,n)
(
i
=
1
,
2
,
⋯
,
n
)
and
b
1
b
2
⋯
b
n
a
1
a
2
⋯
a
n
≤
B
A
.
\frac{b_1 b_2 \cdots b_n}{a_1 a_2 \cdots a_n}\leq \frac{B}{A}.
a
1
a
2
⋯
a
n
b
1
b
2
⋯
b
n
≤
A
B
.
Prove that
(
b
1
+
1
)
(
b
2
+
1
)
⋯
(
b
n
+
1
)
(
a
1
+
1
)
(
a
2
+
1
)
⋯
(
a
n
+
1
)
≤
B
+
1
A
+
1
.
\frac{(b_1+1) (b_2+1) \cdots (b_n+1)}{(a_1+1) (a_2+1) \cdots (a_n+1)}\leq \frac{B+1}{A+1}.
(
a
1
+
1
)
(
a
2
+
1
)
⋯
(
a
n
+
1
)
(
b
1
+
1
)
(
b
2
+
1
)
⋯
(
b
n
+
1
)
≤
A
+
1
B
+
1
.
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