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Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
1994 China Team Selection Test
1
5n numbers inequality
5n numbers inequality
Source: China TST 1994, problem 4
May 17, 2005
inequalities
logarithms
function
algebra
domain
linear algebra
matrix
Problem Statement
Given
5
n
5n
5
n
real numbers
r
i
,
s
i
,
t
i
,
u
i
,
v
i
≥
1
(
1
≤
i
≤
n
)
r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)
r
i
,
s
i
,
t
i
,
u
i
,
v
i
≥
1
(
1
≤
i
≤
n
)
, let
R
=
1
n
∑
i
=
1
n
r
i
R = \frac {1}{n} \sum_{i=1}^{n} r_i
R
=
n
1
∑
i
=
1
n
r
i
,
S
=
1
n
∑
i
=
1
n
s
i
S = \frac {1}{n} \sum_{i=1}^{n} s_i
S
=
n
1
∑
i
=
1
n
s
i
,
T
=
1
n
∑
i
=
1
n
t
i
T = \frac {1}{n} \sum_{i=1}^{n} t_i
T
=
n
1
∑
i
=
1
n
t
i
,
U
=
1
n
∑
i
=
1
n
u
i
U = \frac {1}{n} \sum_{i=1}^{n} u_i
U
=
n
1
∑
i
=
1
n
u
i
,
V
=
1
n
∑
i
=
1
n
v
i
V = \frac {1}{n} \sum_{i=1}^{n} v_i
V
=
n
1
∑
i
=
1
n
v
i
. Prove that
∏
i
=
1
n
r
i
s
i
t
i
u
i
v
i
+
1
r
i
s
i
t
i
u
i
v
i
−
1
≥
(
R
S
T
U
V
+
1
R
S
T
U
V
−
1
)
n
\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n
∏
i
=
1
n
r
i
s
i
t
i
u
i
v
i
−
1
r
i
s
i
t
i
u
i
v
i
+
1
≥
(
RST
U
V
−
1
RST
U
V
+
1
)
n
.
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