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IMO Shortlist
2016 IMO Shortlist
A8
nonstandard inequality
nonstandard inequality
Source: 2016 IMO Shortlist A8
July 19, 2017
IMO Shortlist
inequalities
n-variable inequality
Problem Statement
Find the largest real constant
a
a
a
such that for all
n
≥
1
n \geq 1
n
≥
1
and for all real numbers
x
0
,
x
1
,
.
.
.
,
x
n
x_0, x_1, ... , x_n
x
0
,
x
1
,
...
,
x
n
satisfying
0
=
x
0
<
x
1
<
x
2
<
⋯
<
x
n
0 = x_0 < x_1 < x_2 < \cdots < x_n
0
=
x
0
<
x
1
<
x
2
<
⋯
<
x
n
we have
1
x
1
−
x
0
+
1
x
2
−
x
1
+
⋯
+
1
x
n
−
x
n
−
1
≥
a
(
2
x
1
+
3
x
2
+
⋯
+
n
+
1
x
n
)
\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)
x
1
−
x
0
1
+
x
2
−
x
1
1
+
⋯
+
x
n
−
x
n
−
1
1
≥
a
(
x
1
2
+
x
2
3
+
⋯
+
x
n
n
+
1
)
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