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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2019 Junior Balkan Team Selection Tests - Moldova
2019 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(12)
8
1
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Moldova JTST 2019 P8
It is considered a regular polygon with
n
n
n
sides, where
n
(
n
>
3
)
n(n>3)
n
(
n
>
3
)
is an odd number that does not divide by 3. From the vertices of the polygon are arbitrarily chosen
m
(
0
≤
m
≤
n
)
m(0\leq m\leq n)
m
(
0
≤
m
≤
n
)
vertices that are colored in red and the others in black. A triangle with the vertices at the vertices of the polygon it is considered
m
o
n
o
c
o
l
o
r
monocolor
m
o
n
oco
l
or
,if all of its vertices are of the same color. Prove that the number of all
m
o
n
o
c
o
l
o
r
monocolor
m
o
n
oco
l
or
isosceles triangles with the vertices at the given polygon ends does not depend on the way of coloring of the vertices of the polygon. Determine the number of all these
m
o
n
o
c
o
l
o
r
monocolor
m
o
n
oco
l
or
isosceles triangles.
7
1
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Moldova JTST 2019 P7
Point
H
H
H
is the orthocenter of the acute triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
and point
K
K
K
,situated on the line
(
B
C
)
(BC)
(
BC
)
, is the foot of the perpendicular from point
A
A
A
.The circle
Ω
\Omega
Ω
passes through points
A
A
A
and
K
K
K
,intersecting the sides
(
A
B
)
(AB)
(
A
B
)
and
(
A
C
)
(AC)
(
A
C
)
in points
M
M
M
and
N
N
N
.The line that passes through point
A
A
A
and is parallel with
B
C
BC
BC
intersects again the circumcircles of triangles
Δ
A
H
M
\Delta AHM
Δ
A
H
M
and
Δ
A
H
N
\Delta AHN
Δ
A
H
N
in points
X
X
X
and
Y
Y
Y
.Prove that
X
Y
=
B
C
XY =BC
X
Y
=
BC
.
12
1
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Moldova JTST 2019 P12
The number
B
=
a
1
a
2
…
a
n
a
1
a
2
…
a
n
‾
B=\overline{a_1a_2\dots a_na_1a_2\dots a_n}
B
=
a
1
a
2
…
a
n
a
1
a
2
…
a
n
it is called
r
e
p
e
t
i
t
i
o
n
repetition
re
p
e
t
i
t
i
o
n
of the natural positive number
A
=
a
1
a
2
…
a
n
‾
A = \overline{a_1a_2\dots a_n}
A
=
a
1
a
2
…
a
n
.Prove that there is an infinity of natural numbers ,whose
r
e
p
e
t
i
t
i
o
n
repetition
re
p
e
t
i
t
i
o
n
is a perfect square .
11
1
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Moldova JTST 2019 P11
Let
I
I
I
be the center of inscribed circle of right triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
with
∠
A
=
90
\angle A = 90
∠
A
=
90
and point
M
M
M
is the midpoint of
(
B
C
)
(BC)
(
BC
)
.The bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
intersects the circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
in point
W
W
W
.Point
U
U
U
is situated on the line
A
B
AB
A
B
such that the lines
A
B
AB
A
B
and
W
U
WU
W
U
are perpendiculars.Point
P
P
P
is situated on the line
W
U
WU
W
U
such that the lines
P
I
PI
P
I
and
W
U
WU
W
U
are perpendiculars.Prove that the line
M
P
MP
MP
bisects the segment
C
I
CI
C
I
.
4
1
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Moldova JTST 2019 P4
Let
n
(
n
≥
2
)
n(n\geq2)
n
(
n
≥
2
)
be a natural number and
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
natural positive real numbers. Determine the least possible value of the expression
E
n
=
(
1
+
a
1
)
⋅
(
a
1
+
a
2
)
⋅
(
a
2
+
a
3
)
⋅
.
.
.
⋅
(
a
n
−
1
+
a
n
)
⋅
(
a
n
+
3
n
+
1
)
a
1
⋅
a
2
⋅
a
3
⋅
.
.
.
⋅
a
n
E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n}
E
n
=
a
1
⋅
a
2
⋅
a
3
⋅
...
⋅
a
n
(
1
+
a
1
)
⋅
(
a
1
+
a
2
)
⋅
(
a
2
+
a
3
)
⋅
...
⋅
(
a
n
−
1
+
a
n
)
⋅
(
a
n
+
3
n
+
1
)
10
1
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Moldova JTST 2019 P10
Positive real numbers
a
a
a
and
b
b
b
verify
a
5
+
b
5
=
a
3
+
b
3
a^5+b^5=a^3+b^3
a
5
+
b
5
=
a
3
+
b
3
. Find the greatest possible value of the expression
E
=
a
2
−
a
b
+
b
2
E=a^2-ab+b^2
E
=
a
2
−
ab
+
b
2
.
6
1
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Moldova JTST 2019 P6
Let
p
p
p
and
q
q
q
be integers. If
k
2
+
p
k
+
q
>
0
k^2+pk+q>0
k
2
+
p
k
+
q
>
0
for every integer
k
k
k
, show that
x
2
+
p
x
+
q
>
0
x^2+px+q>0
x
2
+
p
x
+
q
>
0
for every real number
x
x
x
.
2
1
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Moldova JTST 2019 P2
The numeric sequence
(
a
n
)
n
≥
1
(a_n)_{n\geq1}
(
a
n
)
n
≥
1
verifies the relation
a
n
+
1
=
n
+
2
n
⋅
(
a
n
−
1
)
a_{n+1} = \frac{n+2}{n} \cdot (a_n-1)
a
n
+
1
=
n
n
+
2
⋅
(
a
n
−
1
)
for any
n
∈
N
∗
n\in N^*
n
∈
N
∗
.Show that
a
n
∈
Z
a_n \in Z
a
n
∈
Z
for any
n
∈
N
∗
n\in N^*
n
∈
N
∗
,if
a
1
∈
Z
a_1\in Z
a
1
∈
Z
.
9
1
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Moldova JTST 2019 P9
Find all positive real numbers
x
x
x
, that verify
x
+
[
x
3
]
=
[
2
x
3
]
+
[
3
x
5
]
x+\left[\frac{x}{3}\right]=\left[\frac{2x}{3}\right]+\left[\frac{3x}{5}\right]
x
+
[
3
x
]
=
[
3
2
x
]
+
[
5
3
x
]
.
5
1
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Moldova JTST 2019 P5
Find all triplets of positive integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
that verify
(
1
a
+
1
)
(
1
b
+
1
)
(
1
c
+
1
)
=
2
\left(\frac{1}{a}+1\right)\left(\frac{1}{b}+1\right)\left(\frac{1}{c}+1\right)=2
(
a
1
+
1
)
(
b
1
+
1
)
(
c
1
+
1
)
=
2
.
3
1
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Moldova JTST 2019 P3
Let
O
O
O
be the center of circumscribed circle
Ω
\Omega
Ω
of acute triangle
Δ
A
B
C
\Delta ABC
Δ
A
BC
. The line
A
C
AC
A
C
intersects the circumscribed circle of triangle
Δ
A
B
O
\Delta ABO
Δ
A
BO
for the second time in
X
X
X
. Prove that
B
C
BC
BC
and
X
O
XO
XO
are perpendicular.
1
1
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Moldova JTST 2019 P1
Let
n
n
n
be a positive integer. From the set
A
=
{
1
,
2
,
3
,
.
.
.
,
n
}
A=\{1,2,3,...,n\}
A
=
{
1
,
2
,
3
,
...
,
n
}
an element is eliminated. What's the smallest possible cardinal of
A
A
A
and the eliminated element, since the arithmetic mean of left elements in
A
A
A
is
439
13
\frac{439}{13}
13
439
.