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National and Regional Contests
Indonesia Contests
Indonesia TST
2012 Indonesia TST
3
Geometric mean of fractions is larger
Geometric mean of fractions is larger
Source: 2012 Indonesia Round 2 TST 2 Problem 3
March 4, 2012
inequalities unsolved
inequalities
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
,
b
1
,
b
2
,
…
,
b
n
a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n
a
1
,
a
2
,
…
,
a
n
,
b
1
,
b
2
,
…
,
b
n
be positive reals such that
a
1
+
b
1
=
a
2
+
b
2
=
…
+
a
n
+
b
n
a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n
a
1
+
b
1
=
a
2
+
b
2
=
…
+
a
n
+
b
n
and
a
1
a
2
…
a
n
b
1
b
2
…
b
n
n
≥
n
.
\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.
n
b
1
b
2
…
b
n
a
1
a
2
…
a
n
≥
n
.
Prove that
a
1
a
2
…
a
n
b
1
b
2
…
b
n
n
≥
a
1
+
a
2
+
…
+
a
n
b
1
+
b
2
+
…
+
b
n
.
\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.
n
b
1
b
2
…
b
n
a
1
a
2
…
a
n
≥
b
1
+
b
2
+
…
+
b
n
a
1
+
a
2
+
…
+
a
n
.
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