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Bangladesh Contests
Bangladesh Mathematical Olympiad
2023 Bangladesh Mathematical Olympiad
P5
Find the minimum
Find the minimum
Source: BdMO 2023 Secondary National P5
February 12, 2023
algebra
Manipulation
identity
Problem Statement
Let
m
m
m
,
n
n
n
and
p
p
p
are real numbers such that
(
m
+
n
+
p
)
(
1
m
+
1
n
+
1
p
)
=
1
\left(m+n+p\right)\left(\frac 1m + \frac 1n + \frac1p\right) =1
(
m
+
n
+
p
)
(
m
1
+
n
1
+
p
1
)
=
1
. Find all possible values of
1
(
m
+
n
+
p
)
2023
−
1
m
2023
−
1
n
2023
−
1
p
2023
.
\frac 1{(m+n+p)^{2023}} -\frac 1{m^{2023}} -\frac 1{n^{2023}} -\frac 1{p^{2023}}.
(
m
+
n
+
p
)
2023
1
−
m
2023
1
−
n
2023
1
−
p
2023
1
.
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