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International Contests
Czech-Polish-Slovak Match
2001 Czech-Polish-Slovak Match
1
1
Part of
2001 Czech-Polish-Slovak Match
Problems
(1)
Czech-Slovak-Polish match 2001
Source: Titu Andreescu 2000-2001 , page 253
2/22/2004
Let
n
≥
2
n\ge2
n
≥
2
be a natural number, and
a
i
a_i
a
i
be positive numbers, where
i
=
1
,
2
,
⋯
,
n
.
i=1,2,\cdots,n.
i
=
1
,
2
,
⋯
,
n
.
Show that
(
a
1
3
+
1
)
(
a
2
3
+
1
)
⋯
(
a
n
3
+
1
)
≥
(
a
1
2
a
2
+
1
)
(
a
2
2
a
3
+
1
)
⋯
(
a
n
2
a
1
+
1
)
\left(a_1^3+1\right)\left(a_2^3+1\right)\cdots\left(a_n^3+1\right) \geq \left(a_1^2a_2+1\right)\left(a_2^2a_3+1\right)\cdots\left(a_n^2a_1+1\right)
(
a
1
3
+
1
)
(
a
2
3
+
1
)
⋯
(
a
n
3
+
1
)
≥
(
a
1
2
a
2
+
1
)
(
a
2
2
a
3
+
1
)
⋯
(
a
n
2
a
1
+
1
)
inequalities
n-variable inequality
Holder
Convexity