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National and Regional Contests
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All-Russian Olympiad
1986 All Soviet Union Mathematical Olympiad
427
ASU 427 All Soviet Union 1986 1/a1+2/(a1+a2)+...+n/(a1+...+an)<4(1/a1+...+1/an)
ASU 427 All Soviet Union 1986 1/a1+2/(a1+a2)+...+n/(a1+...+an)<4(1/a1+...+1/an)
Source:
August 6, 2019
algebra
inequalities
Problem Statement
Prove that the following inequality holds for all positive
{
a
i
}
\{a_i\}
{
a
i
}
:
1
a
1
+
2
a
1
+
a
2
+
.
.
.
+
n
a
1
+
.
.
.
+
a
n
<
4
(
1
a
1
+
.
.
.
+
1
a
n
)
\frac{1}{a_1} + \frac{2}{a_1+a_2} + ... +\frac{ n}{a_1+...+a_n} < 4\left(\frac{1}{a_1} + ... + \frac{1}{a_n}\right)
a
1
1
+
a
1
+
a
2
2
+
...
+
a
1
+
...
+
a
n
n
<
4
(
a
1
1
+
...
+
a
n
1
)
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