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National and Regional Contests
China Contests
China Team Selection Test
2021 China Team Selection Test
5
Average of consecutive terms at least 1
Average of consecutive terms at least 1
Source: China TST 2021, Test 2, Day 2 P5
March 22, 2021
inequalities
algebra
combinatorics
Problem Statement
Let
n
n
n
be a positive integer and
a
1
,
a
2
,
…
a
2
n
+
1
a_1,a_2,\ldots a_{2n+1}
a
1
,
a
2
,
…
a
2
n
+
1
be positive reals. For
k
=
1
,
2
,
…
,
2
n
+
1
k=1,2,\ldots ,2n+1
k
=
1
,
2
,
…
,
2
n
+
1
, denote
b
k
=
max
0
≤
m
≤
n
(
1
2
m
+
1
∑
i
=
k
−
m
k
+
m
a
i
)
b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right)
b
k
=
max
0
≤
m
≤
n
(
2
m
+
1
1
∑
i
=
k
−
m
k
+
m
a
i
)
, where indices are taken modulo
2
n
+
1
2n+1
2
n
+
1
. Prove that the number of indices
k
k
k
satisfying
b
k
≥
1
b_k\ge 1
b
k
≥
1
does not exceed
2
∑
i
=
1
2
n
+
1
a
i
2\sum_{i=1}^{2n+1} a_i
2
∑
i
=
1
2
n
+
1
a
i
.
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