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Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2008 Junior Balkan Team Selection Tests - Moldova
2008 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(12)
11
1
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isosceles wanted, congurent triangles wanted, parallelograms, AD = BC
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
A
D
=
B
C
,
C
D
∦
A
B
,
A
D
∦
B
C
AD = BC, CD \nparallel AB, AD \nparallel BC
A
D
=
BC
,
C
D
∦
A
B
,
A
D
∦
BC
. Points
M
M
M
and
N
N
N
are the midpoints of the sides
C
D
CD
C
D
and
A
B
AB
A
B
, respectively. a) If
E
E
E
and
F
F
F
are points, such that
M
C
B
F
MCBF
MCBF
and
A
D
M
E
ADME
A
D
ME
are parallelograms, prove that
△
B
F
N
≡
△
A
E
N
\vartriangle BF N \equiv \vartriangle AEN
△
BFN
≡
△
A
EN
. b) Let
P
=
M
N
∩
B
C
P = MN \cap BC
P
=
MN
∩
BC
,
Q
=
A
D
∩
M
N
Q = AD \cap MN
Q
=
A
D
∩
MN
,
R
=
A
D
∩
B
C
R = AD \cap BC
R
=
A
D
∩
BC
. Prove that the triangle
P
Q
R
PQR
PQR
is iscosceles.
7
1
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equilateral triangle criterion with distances, BB_1 = d(B_1, AB) + d(A_1, BC)
In an acute triangle
A
B
C
ABC
A
BC
, points
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
are the midpoints of the sides
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
, respectively. It is known that
A
A
1
=
d
(
A
1
,
A
B
)
+
d
(
A
1
,
A
C
)
AA_1 = d(A_1, AB) + d(A_1, AC)
A
A
1
=
d
(
A
1
,
A
B
)
+
d
(
A
1
,
A
C
)
,
B
B
1
=
d
(
B
1
,
A
B
)
+
d
(
A
1
,
B
C
)
BB1 = d(B_1, AB) + d(A_1, BC)
BB
1
=
d
(
B
1
,
A
B
)
+
d
(
A
1
,
BC
)
,
C
C
1
=
d
(
C
1
,
A
C
)
+
d
(
C
1
,
B
C
)
CC_1 = d(C_1, AC) + d(C_1, BC)
C
C
1
=
d
(
C
1
,
A
C
)
+
d
(
C
1
,
BC
)
, where
d
(
X
,
Y
Z
)
d(X, Y Z)
d
(
X
,
Y
Z
)
denotes the distance from point
X
X
X
to the line
Y
Z
YZ
Y
Z
. Prove, that triangle
A
B
C
ABC
A
BC
is equilateral.
3
1
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< A_1B_1C_1=? , <QOP = 1/2 <B_1C_1D_1 , equal rhombuses ABCD, A_1B_1C_1D_1
Rhombuses
A
B
C
D
ABCD
A
BC
D
and
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
are equal. Side
B
C
BC
BC
intersects sides
B
1
C
1
B_1C_1
B
1
C
1
and
C
1
D
1
C_1D_1
C
1
D
1
at points
M
M
M
and
N
N
N
respectively. Side
A
D
AD
A
D
intersects sides
A
1
B
1
A_1B_1
A
1
B
1
and
A
1
D
1
A_1D_1
A
1
D
1
at points
Q
Q
Q
and
P
P
P
respectively. Let
O
O
O
be the intersection point of lines
M
P
MP
MP
and
Q
N
QN
QN
. Find
∠
A
1
B
1
C
1
\angle A_1B_1C_1
∠
A
1
B
1
C
1
, if
∠
Q
O
P
=
1
2
∠
B
1
C
1
D
1
\angle QOP = \frac12 \angle B_1C_1D_1
∠
QOP
=
2
1
∠
B
1
C
1
D
1
.
2
1
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Easy
BJ2. Positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy inequality \frac {3}{2}\geq a \plus{} b \plus{} c. Find the smallest possible value for S \equal{} abc \plus{} \frac {1}{abc}
8
1
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Islands...
Archipelago consists of
n
n
n
islands :
I
1
,
I
2
,
.
.
.
,
I
n
I_1,I_2,...,I_n
I
1
,
I
2
,
...
,
I
n
and
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
- number of the roads on each island. a_1 \equal{} 55, a_k \equal{} a_{k \minus{} 1} \plus{} (k \minus{} 1), ( k \equal{} 2,3,...,n) a) Does there exist an island with 2008 roads? b) Calculate a_1 \plus{} a_2 \plus{} ... \plus{} a_n.
12
1
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Hiperprime....
Natural nonzero numder, which consists of
m
m
m
digits, is called hiperprime, if its any segment, which consists
1
,
2
,
.
.
.
,
m
1,2,...,m
1
,
2
,
...
,
m
digits is prime (for example
53
53
53
is hiperprime, because numbers
53
,
3
,
5
53,3,5
53
,
3
,
5
are prime). Find all hiperprime numbers.
10
1
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Equation with prime numbers
Solve in prime numbers:
{
2
a
−
b
+
7
c
=
1826
3
a
+
5
b
+
7
c
=
2007
\{\begin{array}{c}\ \ 2a - b + 7c = 1826 \ 3a + 5b + 7c = 2007 \end{array}
{
2
a
−
b
+
7
c
=
1826
3
a
+
5
b
+
7
c
=
2007
9
1
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System of equatins
Find all triplets
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
, that satisfy:
{
x
2
−
2
x
−
4
z
=
3
y
2
−
2
y
−
2
x
=
−
14
z
2
−
4
y
−
4
z
=
−
18
\{\begin{array}{c}\ \ x^2 - 2x - 4z = 3\ y^2 - 2y - 2x = - 14 \ z^2 - 4y - 4z = - 18 \end{array}
{
x
2
−
2
x
−
4
z
=
3
y
2
−
2
y
−
2
x
=
−
14
z
2
−
4
y
−
4
z
=
−
18
5
1
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Find x,y
Find all natural pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
, such that
x
x
x
and
y
y
y
are relative prime and satisfy equality: 2x^2 \plus{} 5xy \plus{} 3y^2 \equal{} 41x \plus{} 62y \plus{} 21.
1
1
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Find triplets...
Find all integers
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
, satisfying equality: x^2(y \minus{} z) \plus{} y^2(z \minus{} x) \plus{} z^2(x \minus{} y) \equal{} 2
4
1
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Table with numbers...
The square table
10
×
10
10\times 10
10
×
10
is divided in squares
1
×
1
1\times1
1
×
1
. In each square
1
×
1
1\times1
1
×
1
is written one of the numers
{
1
,
2
,
3
,
.
.
.
,
9
,
10
}
\{1,2,3,...,9,10\}
{
1
,
2
,
3
,
...
,
9
,
10
}
. Numbers from any two adjacent or diagonally adjacent squares are reciprocal prime. Prove, that there exists a number, which is written in this table at least 17 times.
6
1
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Nice equation
Solve the equation 2(x^2\minus{}3x\plus{}2)\equal{}3 \sqrt{x^3\plus{}8}, where
x
∈
R
x\in R
x
∈
R