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National and Regional Contests
Russia Contests
All-Russian Olympiad
2000 All-Russian Olympiad
4
Sequences and unusual averages
Sequences and unusual averages
Source: All-Russian MO 2000
December 30, 2012
inequalities
algebra unsolved
algebra
Sequences
combinatorics
Problem Statement
Let
a
1
,
a
2
,
⋯
,
a
n
a_1, a_2, \cdots, a_n
a
1
,
a
2
,
⋯
,
a
n
be a sequence of nonnegative integers. For
k
=
1
,
2
,
⋯
,
n
k=1,2,\cdots,n
k
=
1
,
2
,
⋯
,
n
denote
m
k
=
max
1
≤
l
≤
k
a
k
−
l
+
1
+
a
k
−
l
+
2
+
⋯
+
a
k
l
.
m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}.
m
k
=
1
≤
l
≤
k
max
l
a
k
−
l
+
1
+
a
k
−
l
+
2
+
⋯
+
a
k
.
Prove that for every
α
>
0
\alpha > 0
α
>
0
the number of values of
k
k
k
for which
m
k
>
α
m_k > \alpha
m
k
>
α
is less than
a
1
+
a
2
+
⋯
+
a
n
α
.
\frac{a_1+a_2+ \cdots +a_n}{\alpha}.
α
a
1
+
a
2
+
⋯
+
a
n
.
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