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14
Sum of a_i is 1 inequality [ILL 1974]
Sum of a_i is 1 inequality [ILL 1974]
Source:
January 2, 2011
inequalities
inequalities proposed
Problem Statement
Let
n
n
n
and
k
k
k
be natural numbers and
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots ,a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers satisfying
a
1
+
a
2
+
⋯
+
a
n
=
1
a_1+a_2+\cdots +a_n=1
a
1
+
a
2
+
⋯
+
a
n
=
1
. Prove that
1
a
1
k
+
1
a
2
k
+
⋯
+
1
a
n
k
≥
n
k
+
1
.
\dfrac {1} {a_1^{k}}+\dfrac {1} {a_2^{k}}+\cdots +\dfrac {1} {a_n^{k}} \ge n^{k+1}.
a
1
k
1
+
a
2
k
1
+
⋯
+
a
n
k
1
≥
n
k
+
1
.
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