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Tuymaada Olympiad
2000 Tuymaada Olympiad
4
Like vasc's one or cmo i don't remember
Like vasc's one or cmo i don't remember
Source: Tuymaada 2000
January 17, 2006
inequalities
logarithms
inequalities proposed
Problem Statement
Prove for real
x
1
x_1
x
1
,
x
2
x_2
x
2
, .....,
x
n
x_n
x
n
,
0
<
x
k
≤
1
2
0 < x_k \leq {1\over 2}
0
<
x
k
≤
2
1
, the inequality
(
n
x
1
+
⋯
+
x
n
−
1
)
n
≤
(
1
x
1
−
1
)
…
(
1
x
n
−
1
)
.
\left( {n \over x_1 + \dots + x_n} - 1 \right)^n \leq \left( {1 \over x_1} - 1 \right) \dots \left( {1 \over x_n} - 1 \right).
(
x
1
+
⋯
+
x
n
n
−
1
)
n
≤
(
x
1
1
−
1
)
…
(
x
n
1
−
1
)
.
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