MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2013 Junior Balkan Team Selection Tests - Moldova
2013 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
5
1
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pab+qbc+rca <= /8(a + b + c)^2 if p, q, r \in [0,1/2], p + q + r = 1, a,b,c>0
The real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are positive, and the real numbers
p
,
q
,
r
∈
[
0
,
1
/
2
]
p, q, r \in [0,1/2]
p
,
q
,
r
∈
[
0
,
1/2
]
satisfy equality
p
+
q
+
r
=
1
p + q + r = 1
p
+
q
+
r
=
1
. Prove the inequality
p
a
b
+
q
b
c
+
r
c
a
≤
1
8
(
a
+
b
+
c
)
2
.
pab + qbc + rca \le \frac18 (a + b + c)^2.
p
ab
+
q
b
c
+
rc
a
≤
8
1
(
a
+
b
+
c
)
2
.
6
1
Hide problems
4xyz = x^4 + y^4 + z^4 + 1.
Determine all triplets of real numbers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
that satisfy the equation
4
x
y
z
=
x
4
+
y
4
+
z
4
+
1
4xyz = x^4 + y^4 + z^4 + 1
4
x
yz
=
x
4
+
y
4
+
z
4
+
1
.
8
1
Hide problems
rectangle with vertices lattice points of saem coloour, red or blue
A point
M
(
x
,
y
)
M (x, y)
M
(
x
,
y
)
of the Cartesian plane of
x
O
y
xOy
x
O
y
coordinates is called lattice if it has integer coordinates. Each lattice point is colored red or blue. Prove that in the plan there is at least one rectangle with lattice vertices of the same color.
4
1
Hide problems
integer X degrees the angle betweeen hour and minutes
A train from stop
A
A
A
to stop
B
B
B
is traveled in
X
X
X
minutes (
0
<
X
<
60
0 <X <60
0
<
X
<
60
). It is known that when starting from
A
A
A
, as well as when arriving at
B
B
B
, the angle formed by the hour and the minute had measure equal to
X
X
X
degrees. Find
X
X
X
.
2
1
Hide problems
{x \in N | x \in 4a + 7b, a, b \in N}, {x \in N | x\ne 3a + 11b, a, b \in N }
Determine the elements of the sets
A
=
{
x
∈
N
∣
x
≠
4
a
+
7
b
,
a
,
b
∈
N
}
A = \{x \in N | x \ne 4a + 7b, a, b \in N\}
A
=
{
x
∈
N
∣
x
=
4
a
+
7
b
,
a
,
b
∈
N
}
,
B
=
{
x
∈
N
∣
x
≠
3
a
+
11
b
,
a
,
b
∈
N
}
B = \{x \in N | x\ne 3a + 11b, a, b \in N\}
B
=
{
x
∈
N
∣
x
=
3
a
+
11
b
,
a
,
b
∈
N
}
.
7
1
Hide problems
<MDN wanted, MN=MD , square related
The points
M
M
M
and
N
N
N
are located respectively on the diagonal
(
A
C
)
(AC)
(
A
C
)
and the side
(
B
C
)
(BC)
(
BC
)
of the square
A
B
C
D
ABCD
A
BC
D
such that
M
N
=
M
D
MN = MD
MN
=
M
D
. Determine the measure of the angle
M
D
N
MDN
M
D
N
.
3
1
Hide problems
concurrency, orthocenter, midpoints, diameters of circumcircle related
The point
O
O
O
is the center of the circle circumscribed of the acute triangle
A
B
C
ABC
A
BC
, and
H
H
H
is the point of intersection of the heights of this triangle. Let
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
be the points diametrically opposed to the vertices
A
,
B
,
C
A, B , C
A
,
B
,
C
respectively of the triangle, and
A
2
,
B
2
,
C
2
A_2, B_2, C_2
A
2
,
B
2
,
C
2
be the midpoints of the segments
[
A
H
]
,
[
B
H
]
¸
[
C
H
]
[AH], [BH] ¸[CH]
[
A
H
]
,
[
B
H
]
¸
[
C
H
]
respectively . Prove that the lines
A
1
A
2
,
B
1
B
2
,
C
1
C
2
A_1A_2, B_1B_2, C_1C_2
A
1
A
2
,
B
1
B
2
,
C
1
C
2
are concurrent .
1
1
Hide problems
Number Theory
Given are positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
a
a
a
is odd,
b
>
c
b>c
b
>
c
,
a
,
b
,
c
a, b, c
a
,
b
,
c
are coprime and
a
(
b
−
c
)
=
2
b
c
a(b-c) =2bc
a
(
b
−
c
)
=
2
b
c
. Prove that
a
b
c
abc
ab
c
is square