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National and Regional Contests
Japan Contests
Japan TST
2016 Japan TST
7
Inequality with sum of reciprocals
Inequality with sum of reciprocals
Source: Japan TST 2016 P7
January 25, 2021
inequalities
Problem Statement
Find the smallest positive real
k
k
k
such that for any positive integer
n
≥
2
n\ge 2
n
≥
2
and positive reals
a
0
,
a
1
,
…
,
a
n
a_0,a_1,\ldots ,a_n
a
0
,
a
1
,
…
,
a
n
,
1
a
0
+
a
1
+
1
a
0
+
a
1
+
a
2
+
…
+
1
a
0
+
a
1
+
…
+
a
n
<
k
(
1
a
0
+
1
a
1
+
…
+
1
a
n
)
.
\frac{1}{a_0+a_1} +\frac{1}{a_0+a_1+a_2} +\ldots +\frac{1}{a_0+a_1+\ldots +a_n} < k\left(\frac{1}{a_0}+\frac{1}{a_1}+\ldots +\frac{1}{a_n}\right).
a
0
+
a
1
1
+
a
0
+
a
1
+
a
2
1
+
…
+
a
0
+
a
1
+
…
+
a
n
1
<
k
(
a
0
1
+
a
1
1
+
…
+
a
n
1
)
.
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