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South East Mathematical Olympiad
2018 South East Mathematical Olympiad
5
Weird sequence with floor
Weird sequence with floor
Source: 2018 China Southeast MO Grade 10 P5
July 31, 2018
algebra
Problem Statement
Let
{
a
n
}
\{a_n\}
{
a
n
}
be a nonnegative real sequence. Define
X
k
=
∑
i
=
1
2
k
a
i
,
Y
k
=
∑
i
=
1
2
k
⌊
2
k
i
⌋
a
i
,
k
=
0
,
1
,
2
,
.
.
.
X_k = \sum_{i=1}^{2^k}a_i, Y_k = \sum_{i=1}^{2^k}\left\lfloor \frac{2^k}{i}\right\rfloor a_i, k=0,1,2,...
X
k
=
i
=
1
∑
2
k
a
i
,
Y
k
=
i
=
1
∑
2
k
⌊
i
2
k
⌋
a
i
,
k
=
0
,
1
,
2
,
...
Prove that
X
n
≤
Y
n
−
∑
i
=
0
n
−
1
Y
i
≤
∑
i
=
0
n
X
i
X_n\le Y_n - \sum_{i=0}^{n-1} Y_i \le \sum_{i=0}^n X_i
X
n
≤
Y
n
−
∑
i
=
0
n
−
1
Y
i
≤
∑
i
=
0
n
X
i
for all positive integer
n
n
n
. Here
⌊
α
⌋
\lfloor\alpha\rfloor
⌊
α
⌋
denotes the largest integer that does not exceed
α
\alpha
α
.
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