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IMO Longlists
1971 IMO Longlists
18
18
Part of
1971 IMO Longlists
Problems
(1)
Arithmetic and Geometric mean - [IMO LongList 1971]
Source:
1/1/2011
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be positive numbers,
m
g
=
(
a
1
a
2
⋯
a
n
)
n
m_g = \sqrt[n]{(a_1a_2 \cdots a_n)}
m
g
=
n
(
a
1
a
2
⋯
a
n
)
their geometric mean, and
m
a
=
(
a
1
+
a
2
+
⋯
+
a
n
)
n
m_a = \frac{(a_1 + a_2 + \cdots + a_n)}{n}
m
a
=
n
(
a
1
+
a
2
+
⋯
+
a
n
)
their arithmetic mean. Prove that
(
1
+
m
g
)
n
≤
(
1
+
a
1
)
⋯
(
1
+
a
n
)
≤
(
1
+
m
a
)
n
.
(1 + m_g)^n \leq (1 + a_1) \cdots(1 + a_n) \leq (1 + m_a)^n.
(
1
+
m
g
)
n
≤
(
1
+
a
1
)
⋯
(
1
+
a
n
)
≤
(
1
+
m
a
)
n
.
inequalities
inequalities proposed