MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2014 Ukraine Team Selection Test
2
Find the maximum
Find the maximum
Source: Oleksii Klurman , Ukraine
April 25, 2016
inequalities
Problem Statement
Let
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots,x_n
x
1
,
x
2
,
⋯
,
x
n
be postive real numbers such that
x
1
x
2
⋯
x
n
=
1
x_1x_2\cdots x_n=1
x
1
x
2
⋯
x
n
=
1
,
S
=
x
1
3
+
x
2
3
+
⋯
+
x
n
3
S=x^3_1+x^3_2+\cdots+x^3_n
S
=
x
1
3
+
x
2
3
+
⋯
+
x
n
3
.Find the maximum of
x
1
S
−
x
1
3
+
x
1
2
+
x
2
S
−
x
2
3
+
x
2
2
+
⋯
+
x
n
S
−
x
n
3
+
x
n
2
\frac{x_1}{S-x^3_1+x^2_1}+\frac{x_2}{S-x^3_2+x^2_2}+\cdots+\frac{x_n}{S-x^3_n+x^2_n}
S
−
x
1
3
+
x
1
2
x
1
+
S
−
x
2
3
+
x
2
2
x
2
+
⋯
+
S
−
x
n
3
+
x
n
2
x
n
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