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National and Regional Contests
India Contests
India National Olympiad
2020 India National Olympiad
4
Strange Inequality
Strange Inequality
Source: INMO 2020 P4
January 19, 2020
Inequality
inequalities
n-variable inequality
Problem Statement
Let
n
⩾
2
n \geqslant 2
n
⩾
2
be an integer and let
1
<
a
1
≤
a
2
≤
⋯
≤
a
n
1<a_1 \le a_2 \le \dots \le a_n
1
<
a
1
≤
a
2
≤
⋯
≤
a
n
be
n
n
n
real numbers such that
a
1
+
a
2
+
⋯
+
a
n
=
2
n
a_1+a_2+\dots+a_n=2n
a
1
+
a
2
+
⋯
+
a
n
=
2
n
. Prove that
a
1
a
2
…
a
n
−
1
+
a
1
a
2
…
a
n
−
2
+
⋯
+
a
1
a
2
+
a
1
+
2
⩽
a
1
a
2
…
a
n
.
a_1a_2\dots a_{n-1}+a_1a_2\dots a_{n-2}+\dots+a_1a_2+a_1+2 \leqslant a_1a_2\dots a_n.
a
1
a
2
…
a
n
−
1
+
a
1
a
2
…
a
n
−
2
+
⋯
+
a
1
a
2
+
a
1
+
2
⩽
a
1
a
2
…
a
n
.
Proposed by Kapil Pause
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