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Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
1999 Junior Balkan Team Selection Tests - Moldova
1999 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(5)
1
1
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xyz/(x+y+1)= 1998000, xyz/(y+z-1)= 1998000, \frac{xyz}{z+x}= 1998000
Solve in
R
R
R
the system:
{
x
y
z
x
+
y
+
1
=
1998000
x
y
z
y
+
z
−
1
=
1998000
x
y
z
z
+
x
=
1998000
\begin{cases} \dfrac{xyz}{x + y + 1}= 1998000\\ \\ \dfrac{xyz}{y + z - 1}= 1998000 \\ \\ \dfrac{xyz}{z+x}= 1998000 \end{cases}
⎩
⎨
⎧
x
+
y
+
1
x
yz
=
1998000
y
+
z
−
1
x
yz
=
1998000
z
+
x
x
yz
=
1998000
5
1
Hide problems
fractions related to set { 1998/1999 , 1999/2000 }
Let the set
M
=
{
1998
1999
,
1999
2000
}
M =\{\frac{1998}{1999},\frac{1999}{2000} \}
M
=
{
1999
1998
,
2000
1999
}
. The set
M
M
M
is completed with new fractions according to the rule: take two distinct fractions
p
1
q
1
\frac{p_1}{q_1}
q
1
p
1
and
p
2
q
2
\frac{p_2}{q_2}
q
2
p
2
from
M
M
M
thus there are no other numbers in
M
M
M
located between them and a new fraction is formed,
p
1
+
p
2
q
1
+
q
2
\frac{p_1+p_2}{q_1+q_2}
q
1
+
q
2
p
1
+
p
2
which is included in
M
M
M
, etc. Show that, after each procedure, the newly obtained fraction is irreducible and is different from the fractions previously included in
M
M
M
.
4
1
Hide problems
area chasing, starting with equilateral and ratios
Let
A
B
C
ABC
A
BC
be an equilateral triangle of area
1998
1998
1998
cm
2
^2
2
. Points
K
,
L
,
M
K, L, M
K
,
L
,
M
divide the segments
[
A
B
]
,
[
B
C
]
,
[
C
A
]
[AB], [BC] ,[CA]
[
A
B
]
,
[
BC
]
,
[
C
A
]
, respectively, in the ratio
3
:
4
3:4
3
:
4
. Line
A
L
AL
A
L
intersects the lines
C
K
CK
C
K
and
B
M
BM
BM
respectively at the points
P
P
P
and
Q
Q
Q
, and the line
B
M
BM
BM
intersects the line
C
K
CK
C
K
at point
R
R
R
. Find the area of the triangle
P
Q
R
PQR
PQR
.
2
1
Hide problems
sum < AEB + <ADB wanted, ABC right isosceles, EC = 2AE given
Let
A
B
C
ABC
A
BC
be an isosceles right triangle with
∠
A
=
9
0
o
\angle A=90^o
∠
A
=
9
0
o
. Point
D
D
D
is the midpoint of the side
[
A
C
]
[AC]
[
A
C
]
, and point
E
∈
[
A
C
]
E \in [AC]
E
∈
[
A
C
]
is so that
E
C
=
2
A
E
EC = 2AE
EC
=
2
A
E
. Calculate
∠
A
E
B
+
∠
A
D
B
\angle AEB + \angle ADB
∠
A
EB
+
∠
A
D
B
.
3
1
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Number Theory
On the board is written a number with nine non-zero and distinct digits. Prove that we can delete at most seven digits so that the number formed by the digits left to be a perfect square.