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National and Regional Contests
Spain Contests
pOMA and PErA mathematical olympiads
2024 PErA
P3
Surprising n-var ineq
Surprising n-var ineq
Source: PErA 2024/3
March 4, 2024
inequalities
Problem Statement
Let
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dots, x_n
x
1
,
x
2
,
…
,
x
n
be positive real numbers such that
x
1
+
x
2
+
⋯
+
x
n
=
1
x_1+x_2+\cdots + x_n=1
x
1
+
x
2
+
⋯
+
x
n
=
1
. Prove that
∑
i
=
1
n
min
{
x
i
−
1
,
x
i
}
⋅
max
{
x
i
,
x
i
+
1
}
x
i
≤
1
,
\sum_{i=1}^n \frac{\min\{x_{i-1},x_i\}\cdot \max\{x_i,x_{i+1}\}}{x_i}\leq 1,
i
=
1
∑
n
x
i
min
{
x
i
−
1
,
x
i
}
⋅
max
{
x
i
,
x
i
+
1
}
≤
1
,
where we denote
x
0
=
x
n
x_0=x_n
x
0
=
x
n
and
x
n
+
1
=
x
1
x_{n+1}=x_1
x
n
+
1
=
x
1
.
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