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National and Regional Contests
Romania Contests
Romania - Local Contests
Gheorghe Vranceanu
2007 Gheorghe Vranceanu
2007 Gheorghe Vranceanu
Part of
Gheorghe Vranceanu
Subcontests
(5)
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4
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4
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another integral limit involving a sequence
Let be a sequence
(
a
n
)
n
⩾
1
\left( a_n \right)_{n\geqslant 1}
(
a
n
)
n
⩾
1
of real numbers defined recursively as
a
n
=
2007
+
1004
n
2
−
a
n
−
1
−
a
n
−
2
−
⋯
−
a
2
−
a
1
.
a_n=2007+1004n^2-a_{n-1}-a_{n-2}-\cdots -a_2-a_1.
a
n
=
2007
+
1004
n
2
−
a
n
−
1
−
a
n
−
2
−
⋯
−
a
2
−
a
1
.
Calculate:
lim
n
→
∞
1
n
∫
1
a
n
e
1
/
ln
t
d
t
\lim_{n\to\infty} \frac{1}{n}\int_1^{a_n} e^{1/\ln t} dt
n
→
∞
lim
n
1
∫
1
a
n
e
1/
l
n
t
d
t
Limit with inverse of primitive
Let
F
F
F
be the primitive of a continuous function
f
:
R
⟶
(
0
,
∞
)
,
f:\mathbb{R}\longrightarrow (0,\infty ),
f
:
R
⟶
(
0
,
∞
)
,
with
F
(
0
)
=
0.
F(0)=0.
F
(
0
)
=
0.
Determine for which values of
λ
∈
(
0
,
1
)
\lambda \in (0,1)
λ
∈
(
0
,
1
)
the function
(
F
−
1
∘
λ
F
)
/
id.
\left( F^{-1}\circ \lambda F \right)/\text{id.}
(
F
−
1
∘
λ
F
)
/
id.
has limit at
0
,
0,
0
,
and calculate it.
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6
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Very odd problem in set theory involving OSet
For a finite ordered set
X
,
X,
X
,
calculate the fractional part of the number
∑
A
⊂
X
∑
of X belongs to A
the j-th term
2
−
j
.
\sum_{A\subset X} \sum_{\stackrel{\text{the j-th term}}{\text{of X belongs to A}}} 2^{-j} .
∑
A
⊂
X
∑
of X belongs to A
the j-th term
2
−
j
.