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National and Regional Contests
China Contests
XES Mathematics Olympiad
the 12th XMO
Problem 2
Problem 2
Part of
the 12th XMO
Problems
(1)
12th XMO Problem 2
Source: 12th XMO
4/13/2023
Let
a
1
,
a
2
,
⋯
,
a
22
∈
[
1
,
2
]
,
a_1,a_2,\cdots,a_{22}\in [1,2],
a
1
,
a
2
,
⋯
,
a
22
∈
[
1
,
2
]
,
find the maximum value of
∑
i
=
1
22
a
i
a
i
+
1
(
∑
i
=
1
22
a
i
)
2
\dfrac{\sum\limits_{i=1}^{22}a_ia_{i+1}}{\left( \sum\limits_{i=1}^{22}a_i\right) ^2}
(
i
=
1
∑
22
a
i
)
2
i
=
1
∑
22
a
i
a
i
+
1
where
a
23
=
a
1
.
a_{23}=a_1.
a
23
=
a
1
.
xmo
algebra
inequalities