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Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2020 Junior Balkan Team Selection Tests - Moldova
2020 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(12)
6
1
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JBMO TST Moldova Problem 6
The inscribed circle inside triangle
A
B
C
ABC
A
BC
intersects side
A
B
AB
A
B
in
D
D
D
. The inscribed circle inside triangle
A
D
C
ADC
A
D
C
intersects sides
A
D
AD
A
D
in
P
P
P
and
A
C
AC
A
C
in
Q
Q
Q
.The inscribed circle inside triangle
B
D
C
BDC
B
D
C
intersects sides
B
C
BC
BC
in
M
M
M
and
B
D
BD
B
D
in
N
N
N
. Prove that
P
,
Q
,
M
,
N
P , Q, M, N
P
,
Q
,
M
,
N
are cyclic.
5
1
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JBMO TST Moldova Problem 5
Let there be
A
=
1
a
1
2
a
2
…
10
0
a
1
00
A=1^{a_1}2^{a_2}\dots100^{a_100}
A
=
1
a
1
2
a
2
…
10
0
a
1
00
and
B
=
1
b
1
2
b
2
…
10
0
b
1
00
B=1^{b_1}2^{b_2}\dots100^{b_100}
B
=
1
b
1
2
b
2
…
10
0
b
1
00
, where
a
i
,
b
i
∈
N
a_i , b_i \in N
a
i
,
b
i
∈
N
,
a
i
+
b
i
=
101
−
i
a_i + b_i = 101 - i
a
i
+
b
i
=
101
−
i
, (
i
=
1
,
2
,
…
,
100
i= 1,2,\dots,100
i
=
1
,
2
,
…
,
100
). Find the last 1124 digits of
P
=
A
∗
B
P = A * B
P
=
A
∗
B
.
4
1
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JBMO TST Moldova Problem 4
A natural number
n
n
n
is called "
k
k
k
-squared" if it can be written as a sum of
k
k
k
perfect squares not equal to 0.a) Prove that 2020 is "
2
2
2
-squared" , "
3
3
3
-squared" and "
4
4
4
-squared".b) Determine all natural numbers not equal to 0 (
a
,
b
,
c
,
d
,
e
a, b, c, d ,e
a
,
b
,
c
,
d
,
e
)
a
<
b
<
c
<
d
<
e
a<b<c<d<e
a
<
b
<
c
<
d
<
e
that verify the following conditions simultaneously : 1)
e
−
2
e-2
e
−
2
,
e
e
e
,
e
+
4
e+4
e
+
4
are all prime numbers. 2)
a
2
+
b
2
+
c
2
+
d
2
+
e
2
a^2+ b^2 + c^2 + d^2 + e^2
a
2
+
b
2
+
c
2
+
d
2
+
e
2
= 2020.
9
1
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JBMO TST Moldova Problem 9
Find all the real numbers
x
x
x
that verify the equation:
x
−
3
{
x
}
−
{
3
{
x
}
}
=
0
x-3\{x\}-\{3\{x\}\}=0
x
−
3
{
x
}
−
{
3
{
x
}}
=
0
{
a
}
\{a\}
{
a
}
represents the fractional part of
a
a
a
8
1
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JBMO TST Moldova Problem 8
Find the pairs of real numbers
(
a
,
b
)
(a,b)
(
a
,
b
)
such that the biggest of the numbers
x
=
b
2
−
a
−
1
2
x=b^2-\frac{a-1}{2}
x
=
b
2
−
2
a
−
1
and
y
=
a
2
+
b
+
1
2
y=a^2+\frac{b+1}{2}
y
=
a
2
+
2
b
+
1
is less than or equal to
7
16
\frac{7}{16}
16
7
7
1
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JBMO TST Moldova Problem 7
There are written
n
n
n
distinct positive integers. An operation is defined as follows: we chose two numers
a
a
a
and
b
b
b
written on the table; we erase them; we write at their places
a
+
1
a+1
a
+
1
and
b
−
1
b-1
b
−
1
. Find the smallest value of the difference the biggest and the smallest written numbers after some operations.
12
1
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JBMO TST 2020 - Moldova P12
Find all numbers
n
∈
N
∗
n \in \mathbb{N}^*
n
∈
N
∗
for which there exists a finite set of natural numbers
A
=
(
a
1
,
a
2
,
.
.
.
a
n
)
A=(a_1, a_2,...a_n)
A
=
(
a
1
,
a
2
,
...
a
n
)
so that for any
k
k
k
(
1
≤
k
≤
n
)
(1\leq k \leq n)
(
1
≤
k
≤
n
)
the number
a
k
a_k
a
k
is the number of all multiples of
k
k
k
in set
A
A
A
.
11
1
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JBMO TST 2020 - Moldova P11
Let
△
A
B
C
\triangle ABC
△
A
BC
be an acute triangle. The bisector of
∠
A
C
B
\angle ACB
∠
A
CB
intersects side
A
B
AB
A
B
in
D
D
D
. The circumcircle of triangle
A
D
C
ADC
A
D
C
intersects side
B
C
BC
BC
in
C
C
C
and
E
E
E
with
C
≠
E
C \neq E
C
=
E
. The line parallel to
A
E
AE
A
E
which passes through
B
B
B
intersects line
C
D
CD
C
D
in
F
F
F
. Prove that the triangle
△
A
F
B
\triangle AFB
△
A
FB
is isosceles.
3
1
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JBMO TST Moldova Problem 3
Let there be a regular polygon of
n
n
n
sides with center
O
O
O
. Determine the highest possible number of vertices
k
k
k
(
k
≥
3
)
(k \geq 3)
(
k
≥
3
)
, which can be coloured in green, such that
O
O
O
is strictly outside of any triangle with
3
3
3
vertices coloured green. Determine this
k
k
k
for
a
)
n
=
2019
a) n=2019
a
)
n
=
2019
;
b
)
n
=
2020
b) n=2020
b
)
n
=
2020
.
2
1
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JBMO TST Moldova Problem 2
The positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy the equation
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Prove the identity:
(
a
+
b
c
)
(
b
+
c
a
)
c
+
a
b
+
(
b
+
c
a
)
(
c
+
a
b
)
a
+
b
c
+
(
c
+
a
b
)
(
a
+
b
c
)
b
+
c
a
=
2
\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2
c
+
ab
(
a
+
b
c
)
(
b
+
c
a
)
+
a
+
b
c
(
b
+
c
a
)
(
c
+
ab
)
+
b
+
c
a
(
c
+
ab
)
(
a
+
b
c
)
=
2
10
1
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JBMO TST 2020 - Moldova P10
Find all pairs of prime numbers
(
p
,
q
)
(p, q)
(
p
,
q
)
for which the numbers
p
+
q
p+q
p
+
q
and
p
+
4
q
p+4q
p
+
4
q
are simultaneously perfect squares.
1
1
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JBMO TST Moldova Problem 1
Let there be a triangle
A
B
C
ABC
A
BC
with orthocenter
H
H
H
. Let the lengths of the heights be
h
a
,
h
b
,
h
c
h_a, h_b, h_c
h
a
,
h
b
,
h
c
from points
A
,
B
A, B
A
,
B
and respectively
C
C
C
, and the semi-perimeter
p
p
p
of triangle
A
B
C
ABC
A
BC
. It is known that
A
H
⋅
h
a
+
B
H
⋅
h
b
+
C
H
⋅
h
c
=
2
3
⋅
p
2
AH \cdot h_a + BH \cdot h_b + CH \cdot h_c = \frac{2}{3} \cdot p^2
A
H
⋅
h
a
+
B
H
⋅
h
b
+
C
H
⋅
h
c
=
3
2
⋅
p
2
. Show that
A
B
C
ABC
A
BC
is equilateral.