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China National Olympiad
2020 China National Olympiad
1
China Mathematical Olympiad 2020 Q1
China Mathematical Olympiad 2020 Q1
Source: China Mathematical Olympiad 2020 Q1
November 26, 2019
algebra
maximum value
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
41
∈
R
,
a_1,a_2,\cdots,a_{41}\in\mathbb{R},
a
1
,
a
2
,
⋯
,
a
41
∈
R
,
such that
a
41
=
a
1
,
∑
i
=
1
40
a
i
=
0
,
a_{41}=a_1, \sum_{i=1}^{40}a_i=0,
a
41
=
a
1
,
∑
i
=
1
40
a
i
=
0
,
and for any
i
=
1
,
2
,
⋯
,
40
,
∣
a
i
−
a
i
+
1
∣
≤
1.
i=1,2,\cdots,40, |a_i-a_{i+1}|\leq 1.
i
=
1
,
2
,
⋯
,
40
,
∣
a
i
−
a
i
+
1
∣
≤
1.
Determine the greatest possible value of
(
1
)
a
10
+
a
20
+
a
30
+
a
40
;
(1)a_{10}+a_{20}+a_{30}+a_{40};
(
1
)
a
10
+
a
20
+
a
30
+
a
40
;
(
2
)
a
10
⋅
a
20
+
a
30
⋅
a
40
.
(2)a_{10}\cdot a_{20}+a_{30}\cdot a_{40}.
(
2
)
a
10
⋅
a
20
+
a
30
⋅
a
40
.
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