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Sum of k-th powers - JBMO Shortlist
Sum of k-th powers - JBMO Shortlist
Source:
October 30, 2010
inequalities proposed
inequalities
Problem Statement
Prove that
(
1
k
+
2
k
)
(
1
k
+
2
k
+
3
k
)
…
(
1
k
+
2
k
+
…
+
n
k
)
\sqrt{(1^k+2^k)(1^k+2^k+3^k)\ldots (1^k+2^k+\ldots +n^k)}
(
1
k
+
2
k
)
(
1
k
+
2
k
+
3
k
)
…
(
1
k
+
2
k
+
…
+
n
k
)
≥
1
k
+
2
k
+
…
+
n
k
−
2
k
−
1
+
2
⋅
3
k
−
1
+
…
+
(
n
−
1
)
⋅
n
k
−
1
n
\ge 1^k+2^k+\ldots +n^k-\frac{2^{k-1}+2\cdot 3^{k-1}+\ldots + (n-1)\cdot n^{k-1}}{n}
≥
1
k
+
2
k
+
…
+
n
k
−
n
2
k
−
1
+
2
⋅
3
k
−
1
+
…
+
(
n
−
1
)
⋅
n
k
−
1
for all integers
n
,
k
≥
2
n,k \ge 2
n
,
k
≥
2
.
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