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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2015 Junior Balkan Team Selection Tests - Moldova
2015 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
8
1
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lcm, [a, b] =1000, [b, c] = 2000, [c, a] =2000
Determine the number of all ordered triplets of positive integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
, which satisfy the equalities:
[
a
,
b
]
=
1000
,
[
b
,
c
]
=
2000
,
[
c
,
a
]
=
2000.
[a, b] =1000, [b, c] = 2000, [c, a] =2000.
[
a
,
b
]
=
1000
,
[
b
,
c
]
=
2000
,
[
c
,
a
]
=
2000.
([x, y]represents the least common multiple of positive integers x,y)
6
1
Hide problems
E=a^{2015}+b^{2015}+c^{2015}. if 2015 (a + b + c) =1, ab+bc+ca=2015abc
Real numbers
a
,
b
a,b
a
,
b
and
c
c
c
satisfy the equalities
2015
(
a
+
b
+
c
)
=
1
2015 (a + b + c) =1
2015
(
a
+
b
+
c
)
=
1
and
a
b
+
b
c
+
c
a
=
2015
a
b
c
ab+bc+ca=2015 abc
ab
+
b
c
+
c
a
=
2015
ab
c
. Determine the numeric value of the expression
E
=
a
2015
+
b
2015
+
c
2015
.
E=a^{2015}+b^{2015}+c^{2015}.
E
=
a
2015
+
b
2015
+
c
2015
.
5
1
Hide problems
find 19 other elements of set of integers A
The set A contains exactly
21
21
21
integers. The sum of any
11
11
11
numbers in
A
A
A
is greater than the sum of the remaining numbers. It is known that the set
A
A
A
contain thes number
101
101
101
, and the largest number in
A
A
A
is
2014
2014
2014
. Find out the other
19
19
19
numbers in
A
A
A
.
4
1
Hide problems
game with numbers 1, 2,. . . , 33 are written on the board
The numbers
1
,
2
,
.
.
.
,
33
1, 2,. . . , 33
1
,
2
,
...
,
33
are written on the board . A student performs the following procedure: choose two numbers from those written on the board so that one of them is a multiple of the other number; after the election he deletes the two numbers and writes on the board their number. The student repeats the procedure so many times until only numbers without multiples remain on the board. Determine how many numbers they remain on the board in the situation where the student can no longer repeat the procedure.
1
1
Hide problems
compare 2014^{9^{9^a}}, 2015^{a^{a^9}} if a=123456789
Ler
a
a
a
be the number
123456789
123456789
123456789
. Compare the numbers
201
4
9
9
a
,
201
5
a
a
9
2014^{9^{9^a}}, 2015^{a^{a^9}}
201
4
9
9
a
,
201
5
a
a
9
2
1
Hide problems
product of 15 increased numbers by is 2015 their product
Show an example of
15
15
15
nonzero natural numbers with the property that if each one of them is increased by one, then the product of all increased numbers is
2015
2015
2015
times higher than the product of the initial numbers
7
1
Hide problems
AB= CE wanted, starting with a 90-54-36 triangle
In a right triangle
A
B
C
ABC
A
BC
with
∠
B
A
C
=
9
0
o
\angle BAC =90^o
∠
B
A
C
=
9
0
o
and
∠
A
B
C
=
5
4
o
\angle ABC= 54^o
∠
A
BC
=
5
4
o
, point
M
M
M
is the midpoint of the hypotenuse
[
B
C
]
[BC]
[
BC
]
, point
D
D
D
is the foot of the angle bisector drawn from the vertex
C
C
C
and
A
M
∩
C
D
=
{
E
}
AM \cap CD = \{E\}
A
M
∩
C
D
=
{
E
}
. Prove that
A
B
=
C
E
AB= CE
A
B
=
CE
.
3
1
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parallel wanted, tangents at circumcircle related
Let
Ω
\Omega
Ω
be the circle circumscribed to the triangle
A
B
C
ABC
A
BC
. Tangents taken to the circle
Ω
\Omega
Ω
at points
A
A
A
and
B
B
B
intersects at the point
P
P
P
, and the perpendicular bisector of
(
B
C
)
(BC)
(
BC
)
cuts line
A
C
AC
A
C
at point
Q
Q
Q
. Prove that lines
B
C
BC
BC
and
P
Q
PQ
PQ
are parallel.