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International Contests
CentroAmerican
2021 Centroamerican and Caribbean Math Olympiad
5
5
Part of
2021 Centroamerican and Caribbean Math Olympiad
Problems
(1)
Inequality with strange square condition
Source: Cetroamerican 2021
8/12/2021
Let
n
≥
3
n \geq 3
n
≥
3
be an integer and
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
be positive real numbers such that
m
m
m
is the smallest and
M
M
M
is the largest of these numbers. It is known that for any distinct integers
1
≤
i
,
j
,
k
≤
n
1 \leq i,j,k \leq n
1
≤
i
,
j
,
k
≤
n
, if
a
i
≤
a
j
≤
a
k
a_i \leq a_j \leq a_k
a
i
≤
a
j
≤
a
k
then
a
i
a
k
≤
a
j
2
a_ia_k \leq a_j^2
a
i
a
k
≤
a
j
2
. Show that
a
1
a
2
⋯
a
n
≥
m
2
M
n
−
2
a_1a_2 \cdots a_n \geq m^2M^{n-2}
a
1
a
2
⋯
a
n
≥
m
2
M
n
−
2
and determine when equality holds
inequalities