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Contest of the Centers of Excellency of Suceava
2019 Centers of Excellency of Suceava
2019 Centers of Excellency of Suceava
Part of
Contest of the Centers of Excellency of Suceava
Subcontests
(3)
3
3
Hide problems
Geometric equality
The circumcenter, circumradius and orthocenter of a triangle
A
B
C
ABC
A
BC
satisfying
A
B
<
A
C
AB<AC
A
B
<
A
C
are notated with
O
,
R
,
H
,
O,R,H,
O
,
R
,
H
,
respectively. Prove that the middle of the segment
O
H
OH
O
H
belongs to the line
B
C
BC
BC
if
A
C
2
−
A
B
2
=
2
R
⋅
B
C
.
AC^2-AB^2=2R\cdot BC.
A
C
2
−
A
B
2
=
2
R
⋅
BC
.
Marius Marchitan
On the relation between some arithmetic and geometric progressions
Let
(
a
n
)
n
≥
1
\left( a_n \right)_{n\ge 1}
(
a
n
)
n
≥
1
be a non-constant arithmetic progression of positive numbers and
(
g
n
)
n
≥
1
\left( g_n \right)_{n\ge 1}
(
g
n
)
n
≥
1
be a non-constant geometric progression of positive numbers satisfying
a
1
=
g
1
a_1=g_1
a
1
=
g
1
and
a
2019
=
g
2019
.
a_{2019} =g_{2019} .
a
2019
=
g
2019
.
Specify the set
{
k
∈
N
∣
a
k
≤
g
k
}
\left\{ k\in\mathbb{N} \big| a_k\le g_k \right\}
{
k
∈
N
a
k
≤
g
k
}
and prove that it bijects the natural numbers. Gheorghe Rotariu
On functions f,g satisfying (fg)'=f'g'
For two real intervals
I
,
J
,
I,J,
I
,
J
,
we say that two functions
f
,
g
:
I
⟶
J
f,g:I\longrightarrow J
f
,
g
:
I
⟶
J
have property
P
\mathcal{P}
P
if they are differentiable and
(
f
g
)
′
=
f
′
g
′
.
(fg)'=f'g'.
(
f
g
)
′
=
f
′
g
′
.
a) Provide example of two nonconstant functions
a
,
b
:
R
⟶
R
a,b:\mathbb{R}\longrightarrow\mathbb{R}
a
,
b
:
R
⟶
R
that have property
P
.
\mathcal{P} .
P
.
b) Find the functions
λ
:
(
2019
,
∞
)
⟶
(
0
,
∞
)
\lambda :(2019,\infty )\longrightarrow (0,\infty )
λ
:
(
2019
,
∞
)
⟶
(
0
,
∞
)
having the property that
λ
\lambda
λ
along with
θ
:
(
2019
,
∞
)
⟶
(
0
,
∞
)
,
θ
(
x
)
=
x
2019
\theta :(2019,\infty )\longrightarrow (0,\infty ), \theta (x)=x^{2019}
θ
:
(
2019
,
∞
)
⟶
(
0
,
∞
)
,
θ
(
x
)
=
x
2019
have property
P
.
\mathcal{P} .
P
.
Dan Nedeianu
2
3
Hide problems
Floor of an expression involving n-th square root
For a natural number
n
≥
2
,
n\ge 2,
n
≥
2
,
calculate the integer part of
1
+
n
n
−
2
/
n
.
\sqrt[n]{1+n}-\sqrt {2/n} .
n
1
+
n
−
2/
n
.
Dan Nedeianu
Known prime generating sequence has coprime terms
Let
(
s
n
)
n
≥
1
\left( s_n \right)_{n\ge 1 }
(
s
n
)
n
≥
1
be a sequence with
s
1
s_1
s
1
and defined recursively as
s
n
+
1
=
s
n
2
−
s
n
+
1.
s_{n+1}=s_n^2-s_n+1.
s
n
+
1
=
s
n
2
−
s
n
+
1.
Prove that any two terms of this sequence are coprime. Dan Nedeianu
Limit of sequence defined as a second order inequality recursion
Let be two real numbers
b
>
a
>
0
,
b>a>0,
b
>
a
>
0
,
and a sequence
(
x
n
)
n
≥
1
\left( x_n \right)_{n\ge 1}
(
x
n
)
n
≥
1
with
x
2
>
x
1
>
0
x_2>x_1>0
x
2
>
x
1
>
0
and such that
a
x
n
+
2
+
b
x
n
≥
(
a
+
b
)
x
n
+
1
,
ax_{n+2}+bx_n\ge (a+b)x_{n+1} ,
a
x
n
+
2
+
b
x
n
≥
(
a
+
b
)
x
n
+
1
,
for any natural numbers
n
.
n.
n
.
Prove that
lim
n
→
∞
x
n
=
∞
.
\lim_{n\to\infty } x_n=\infty .
lim
n
→
∞
x
n
=
∞.
Dan Popescu
1
3
Hide problems
Four term - inequality
For
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
positive, prove:
2
a
a
2
+
b
c
+
2
b
b
2
+
c
d
+
2
c
c
2
+
d
a
+
2
d
d
2
+
a
b
≤
1
a
+
1
b
+
1
c
+
1
d
\frac{2a}{a^2+bc} +\frac{2b}{b^2+cd} +\frac{2c}{c^2+da} +\frac{2d}{d^2+ab}\le \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d}
a
2
+
b
c
2
a
+
b
2
+
c
d
2
b
+
c
2
+
d
a
2
c
+
d
2
+
ab
2
d
≤
a
1
+
b
1
+
c
1
+
d
1
Dan Popescu
Combinatoric AM-GM
Prove that
(
m
+
n
min
(
m
,
n
)
)
≤
(
2
m
m
)
⋅
(
2
n
n
)
,
\binom{m+n}{\min (m,n)}\le \sqrt{\binom{2m}{m}\cdot \binom{2n}{n}} ,
(
m
i
n
(
m
,
n
)
m
+
n
)
≤
(
m
2
m
)
⋅
(
n
2
n
)
,
for nonnegative
m
,
n
.
m,n.
m
,
n
.
Gheorghe Stoica
sum of four squares prime
Prove that if a prime is the sum of four perfect squares then the product of two of these is equal to the product of the other two. Gherghe Stoica