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A5
Riemann Sums
Riemann Sums
Source: 2021 ISL A5
July 12, 2022
algebra
real analysis
inequalities
Problem Statement
Let
n
≥
2
n\geq 2
n
≥
2
be an integer and let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be positive real numbers with sum
1
1
1
. Prove that
∑
k
=
1
n
a
k
1
−
a
k
(
a
1
+
a
2
+
⋯
+
a
k
−
1
)
2
<
1
3
.
\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.
k
=
1
∑
n
1
−
a
k
a
k
(
a
1
+
a
2
+
⋯
+
a
k
−
1
)
2
<
3
1
.
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