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Vietnam Contests
Vietnam National Olympiad
1981 Vietnam National Olympiad
2
A^2/B >= 4pq/(p+q)^2.
A^2/B >= 4pq/(p+q)^2.
Source: Vietnam MO 1981 P5
March 17, 2011
inequalities
inequalities unsolved
Problem Statement
Let
p
,
q
p, q
p
,
q
be real numbers with
0
<
p
<
q
0 < p < q
0
<
p
<
q
and let
t
1
,
t
2
,
⋯
,
t
n
t_1, t_2, \cdots, t_n
t
1
,
t
2
,
⋯
,
t
n
be real numbers in the interval
[
p
,
q
]
[p, q]
[
p
,
q
]
. Denote by
A
A
A
and
B
B
B
the arithmetic means of
t
1
,
t
2
,
⋯
,
t
n
t_1, t_2, \cdots, t_n
t
1
,
t
2
,
⋯
,
t
n
and of
t
1
2
,
t
2
2
,
⋯
,
t
n
2
t_1^2, t_2^2,\cdots , t_n^2
t
1
2
,
t
2
2
,
⋯
,
t
n
2
, respectively. Prove that
A
2
B
≥
4
p
q
(
p
+
q
)
2
.
\frac{A^2}{B}\ge\frac{4pq}{(p + q)^2}.
B
A
2
≥
(
p
+
q
)
2
4
pq
.
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