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China Girls Math Olympiad
2024 China Girls Math Olympiad
7
n-variable inequality in CGMO
n-variable inequality in CGMO
Source: CGMO 2024 P7
August 14, 2024
inequalities
Problem Statement
Let
n
n
n
be a positive integer. If
x
1
,
x
2
,
…
,
x
n
≥
0
x_1, x_2, \ldots, x_n \geq 0
x
1
,
x
2
,
…
,
x
n
≥
0
,
x
1
+
x
2
+
…
+
x
n
=
1
x_1+x_2+\ldots+x_n=1
x
1
+
x
2
+
…
+
x
n
=
1
and, assuming
x
n
+
1
=
x
1
x_{n+1}=x_1
x
n
+
1
=
x
1
, find the maximal value of
∑
k
=
1
n
1
+
x
k
2
+
x
k
4
1
+
x
k
+
1
+
x
k
+
1
2
+
x
k
+
1
3
+
x
k
+
1
4
.
\sum_{k=1}^n \frac{1+x_k^2+x_k^4}{1+x_{k+1}+x_{k+1}^2+x_{k+1}^3+x_{k+1}^4}.
k
=
1
∑
n
1
+
x
k
+
1
+
x
k
+
1
2
+
x
k
+
1
3
+
x
k
+
1
4
1
+
x
k
2
+
x
k
4
.
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