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Moldova National Olympiad
2006 Moldova National Olympiad
12.2
Integral with floor function
Integral with floor function
Source:
February 7, 2019
calculus
integration
limits
algebra
floor function
function
Problem Statement
Let
a
,
b
,
n
∈
N
a, b, n \in \mathbb{N}
a
,
b
,
n
∈
N
, with
a
,
b
≥
2.
a, b \geq 2.
a
,
b
≥
2.
Also, let
I
1
(
n
)
=
∫
0
1
⌊
a
n
x
⌋
d
x
I_{1}(n)=\int_{0}^{1} \left \lfloor{a^n x} \right \rfloor dx
I
1
(
n
)
=
∫
0
1
⌊
a
n
x
⌋
d
x
and
I
2
(
n
)
=
∫
0
1
⌊
b
n
x
⌋
d
x
.
I_{2} (n) = \int_{0}^{1} \left \lfloor{b^n x} \right \rfloor dx.
I
2
(
n
)
=
∫
0
1
⌊
b
n
x
⌋
d
x
.
Find
lim
n
→
∞
I
1
(
n
)
I
2
(
n
)
.
\lim_{n \to \infty} \dfrac{I_1(n)}{I_{2}(n)}.
lim
n
→
∞
I
2
(
n
)
I
1
(
n
)
.
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