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National and Regional Contests
Singapore Contests
Singapore MO Open
2024 Singapore MO Open
Q2
Classic inequality on distinct positive integers
Classic inequality on distinct positive integers
Source: SMO open 2024 Q2
August 3, 2024
inequalities
Problem Statement
Let
n
n
n
be a fixed positive integer. Find the minimum value of
x
1
3
+
⋯
+
x
n
3
x
1
+
⋯
+
x
n
\frac{x_1^3+\dots+x_n^3}{x_1+\dots+x_n}
x
1
+
⋯
+
x
n
x
1
3
+
⋯
+
x
n
3
where
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots,x_n
x
1
,
x
2
,
…
,
x
n
are distinct positive integers.
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