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2020 Moldova Team Selection Test
10
Find the greatest value
Find the greatest value
Source: Moldova TST 2020
March 8, 2020
maximum value
algebra
inequalities
Moldova
Problem Statement
Let
n
n
n
be a positive integer. Positive numbers
a
a
a
,
b
b
b
,
c
c
c
satisfy
1
a
+
1
b
+
1
c
=
1
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1
a
1
+
b
1
+
c
1
=
1
. Find the greatest possible value of
E
(
a
,
b
,
c
)
=
a
n
a
2
n
+
1
+
b
2
n
⋅
c
+
b
⋅
c
2
n
+
b
n
b
2
n
+
1
+
c
2
n
⋅
a
+
c
⋅
a
2
n
+
c
n
c
2
n
+
1
+
a
2
n
⋅
b
+
a
⋅
b
2
n
E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}
E
(
a
,
b
,
c
)
=
a
2
n
+
1
+
b
2
n
⋅
c
+
b
⋅
c
2
n
a
n
+
b
2
n
+
1
+
c
2
n
⋅
a
+
c
⋅
a
2
n
b
n
+
c
2
n
+
1
+
a
2
n
⋅
b
+
a
⋅
b
2
n
c
n
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