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Contests
National and Regional Contests
Romania Contests
Romania - Local Contests
Gheorghe Vranceanu
2004 Gheorghe Vranceanu
2004 Gheorghe Vranceanu
Part of
Gheorghe Vranceanu
Subcontests
(5)
4
4
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3
3
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Roots of parametric f, and inverses
Consider the function
f
:
(
−
∞
,
1
]
⟶
R
f:(-\infty,1]\longrightarrow\mathbb{R}
f
:
(
−
∞
,
1
]
⟶
R
defined as
f
(
x
)
=
{
5
2
+
2
x
−
1
2
x
,
e
m
s
p
;
x
<
−
1
3
1
−
x
2
,
e
m
s
p
;
x
∈
[
−
1
,
1
]
.
f(x)=\left\{ \begin{matrix} \frac{5}{2} +2^x-\frac{1}{2^x} ,&   x<-1 \\ 3^{\sqrt{1-x^2}} ,&   x\in [-1,1] \end{matrix} \right. .
f
(
x
)
=
{
2
5
+
2
x
−
2
x
1
,
3
1
−
x
2
,
e
m
s
p
;
x
<
−
1
e
m
s
p
;
x
∈
[
−
1
,
1
]
.
a) For a fixed parameter, find the roots of
f
−
m
.
f-m.
f
−
m
.
b) Study the inversability of the restrictions of
f
f
f
to
(
−
∞
,
−
1
]
(-\infty,-1]
(
−
∞
,
−
1
]
and
[
−
1
,
1
]
[-1,1]
[
−
1
,
1
]
and find the inverses of these that admit them. D. Zaharia
Inequality: floor, modulus and maximum
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be real numbers satisfying
⌊
a
2
+
b
2
+
c
2
⌋
≤
⌊
a
b
+
b
c
+
c
a
⌋
.
\left\lfloor a^2+b^2+c^2 \right\rfloor \le\lfloor ab+bc+ca \rfloor .
⌊
a
2
+
b
2
+
c
2
⌋
≤
⌊
ab
+
b
c
+
c
a
⌋
.
Show that:
2
>
max
{
∣
−
2
a
+
b
+
c
∣
,
∣
a
−
2
b
+
c
∣
,
∣
a
+
b
−
2
c
∣
}
2 >\max\left\{ \left| -2a+b+c \right| ,\left| a-2b+c \right| ,\left| a+b-2c \right| \right\}
2
>
max
{
∣
−
2
a
+
b
+
c
∣
,
∣
a
−
2
b
+
c
∣
,
∣
a
+
b
−
2
c
∣
}
Merticaru
Closedness property of primitives under sufficient conditions
Let be a real number
r
r
r
and two functions
f
:
[
r
,
∞
)
⟶
R
,
F
1
:
(
r
,
∞
)
⟶
R
f:[r,\infty )\longrightarrow\mathbb{R} , F_1:(r,\infty )\longrightarrow\mathbb{R}
f
:
[
r
,
∞
)
⟶
R
,
F
1
:
(
r
,
∞
)
⟶
R
satisfying the following two properties.
(i)
f
\text{(i)} f
(i)
f
has Darboux's intermediate value property.
(ii)
F
1
\text{(ii)} F_1
(ii)
F
1
is differentiable and
F
1
′
=
f
∣
(
r
,
∞
)
F'_1=f\bigg|_{(r,\infty )}
F
1
′
=
f
(
r
,
∞
)
1) Provide an example of what
f
,
F
1
f,F_1
f
,
F
1
could be if
f
f
f
hasn't a lateral limit at
r
,
r,
r
,
and
F
1
F_1
F
1
has lateral limit at
r
.
r.
r
.
Moreover, if
f
f
f
has lateral limit at
r
,
r,
r
,
show that2)
F
1
F_1
F
1
has a finite lateral limit at
r
.
r.
r
.
3) the function
F
:
[
r
,
∞
)
⟶
R
F:[r,\infty )\longrightarrow\mathbb{R}
F
:
[
r
,
∞
)
⟶
R
defined as
F
(
x
)
=
{
F
1
(
x
)
,
e
m
s
p
;
x
∈
(
r
,
∞
)
lim
x
>
r
x
→
r
F
1
(
x
)
,
e
m
s
p
;
x
=
r
F(x)=\left\{ \begin{matrix} F_1(x) ,&   x\in (r,\infty ) \\ \lim_{\stackrel{x\to r}{x>r}} F_1(x), &   x=r \end{matrix} \right.
F
(
x
)
=
{
F
1
(
x
)
,
lim
x
>
r
x
→
r
F
1
(
x
)
,
e
m
s
p
;
x
∈
(
r
,
∞
)
e
m
s
p
;
x
=
r
is a primitive of
f
.
f.
f
.
2
4
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Natural sequence; analysis
How many sequences
(
a
n
)
n
≥
1
\left( a_n \right)_{n\ge 1}
(
a
n
)
n
≥
1
satisfying the following properties do there are?
(1)
\text{(1)}
(1)
All terms of
(
a
n
)
n
≥
1
\left( a_n \right)_{n\ge 1}
(
a
n
)
n
≥
1
are natural numbers.
(2)
(
a
n
)
n
≥
1
\text{(2)} \left( a_n \right)_{n\ge 1}
(2)
(
a
n
)
n
≥
1
is strictly increasing.
(3)
1
a
n
+
1
<
ln
(
1
+
n
)
−
ln
n
<
1
a
n
,
\text{(3)} \frac{1}{a_{n+1}} <\ln (1+n) -\ln n <\frac{1}{a_n} ,
(3)
a
n
+
1
1
<
ln
(
1
+
n
)
−
ln
n
<
a
n
1
,
for any natural number
n
.
n.
n
.
1
4
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