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Problems
Contests
National and Regional Contests
Moldova Contests
Moldova National Olympiad
2009 Moldova National Olympiad
2009 Moldova National Olympiad
Part of
Moldova National Olympiad
Subcontests
(10)
10.4
1
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AP_|_BN wanted, if MP= PC sin^2 C, isosceles ABC
Let the isosceles triangle
A
B
C
ABC
A
BC
with
∣
A
B
∣
=
∣
A
C
∣
| AB | = | AC |
∣
A
B
∣
=
∣
A
C
∣
. The point
M
M
M
is the midpoint of the base
[
B
C
]
[BC]
[
BC
]
, the point
N
N
N
is the orthogonal projection of the point
M
M
M
on the line
A
C
AC
A
C
, and the point
P
P
P
is located on the segment
(
M
C
)
(MC)
(
MC
)
such that
∣
M
P
∣
=
∣
P
C
∣
sin
2
C
| MP | = | P C | \sin^2 C
∣
MP
∣
=
∣
PC
∣
sin
2
C
. Prove that the lines
A
P
AP
A
P
and
B
N
BN
BN
are perpendicular.
10.3
1
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A, B, M, I concyclic if AI=MI where I incenter , M midpoint of BC
Let the triangle
A
B
C
ABC
A
BC
be with
∣
A
B
∣
>
∣
A
C
∣
| AB | > | AC |
∣
A
B
∣
>
∣
A
C
∣
. Point M is the midpoint of the side
[
B
C
]
[BC]
[
BC
]
, and point
I
I
I
is the center of the circle inscribed in the triangle ABC such that the relation
∣
A
I
∣
=
∣
M
I
∣
| AI | = | MI |
∣
A
I
∣
=
∣
M
I
∣
. Prove that points
A
,
B
,
M
,
I
A, B, M, I
A
,
B
,
M
,
I
are located on the same circle.
9.4
1
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cut a triangle into at least 2 triangles with each side=1, when it has side >1
A side of an arbitrary triangle has a length greater than
1
1
1
. Prove that the given triangle it can be cut into at least
2
2
2
triangles, so that each of them has a side of length equal to
1
1
1
.
9.3
1
Hide problems
AC // BK wanted, equilaterals ABC and MKC
Let
A
B
C
ABC
A
BC
be an equilateral triangle. The points
M
M
M
and
K
K
K
are located in different half-planes with respect to line
B
C
BC
BC
, so that the point
M
∈
(
A
B
)
M \in (AB)
M
∈
(
A
B
)
¸and the triangle
M
K
C
MKC
M
K
C
is equilateral. Prove that the lines
A
C
AC
A
C
and
B
K
BK
B
K
are parallel.
8.4
1
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right triangle with 30^o, criterion, incenter related (2009 Moldova NMO 8.4)
Prove that a right triangle has an angle equal to
3
0
o
30^o
3
0
o
if and only if the center of the circle inscribed in this triangle is located on the perpendicular bisector of the median taken from the vertex of the right angle of the triangle.
8.3
1
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\\ wanted, intersecting circles, tangents (2009 Moldova NMO 8.3)
The circle
C
1
C_1
C
1
of center
O
O
O
and the circle
C
2
C_2
C
2
intersect at points
A
A
A
and
B
B
B
, so that point
O
O
O
lies on circle
C
2
C_2
C
2
. The lines
d
d
d
and
e
e
e
are tangent at point
A
A
A
to the circles
C
1
C_1
C
1
and
C
2
C_2
C
2
respectively. If the line
e
e
e
intersects the circle
C
1
C_1
C
1
at point
D
D
D
, prove that the lines
B
D
BD
B
D
and
d
d
d
are parallel.
7.3
1
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2009 points on line AB not in [AB], diff. sums (2009 Moldova NMO 7.3)
On the lines
A
B
AB
A
B
are located
2009
2009
2009
different points that do not belong to the segment
[
A
B
]
[AB]
[
A
B
]
. Prove that the sum of the distances from point
A
A
A
to these points is not equal to the sum of the distances from point
B
B
B
to these points.
7.4
1
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computational in 90-75-15 triangle (2009 Moldova NMO 7.4)
Triangle
A
B
C
ABC
A
BC
with
A
B
=
10
AB = 10
A
B
=
10
cm ¸and
∠
C
=
1
5
o
\angle C= 15^o
∠
C
=
1
5
o
, is right at
B
B
B
. Point
D
∈
(
A
C
)
D \in (AC)
D
∈
(
A
C
)
is the foot of the altitude taken from
B
B
B
. Find the distance from point
D
D
D
to the line
A
B
AB
A
B
.
12.3
1
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Trigonometric equation
Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
of real numbers, so that
sin
(
2009
x
)
+
sin
(
a
x
)
+
sin
(
b
x
)
=
0
\sin(2009x)+\sin(ax)+\sin(bx)=0
sin
(
2009
x
)
+
sin
(
a
x
)
+
sin
(
b
x
)
=
0
holds for any
x
∈
R
x\in \mathbf {R}
x
∈
R
.
12.1
1
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Calculate definite integral
Calculate
∫
−
π
2
π
2
c
o
s
(
x
)
7
e
x
+
1
d
x
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{cos(x)^7}{e^x+1} dx
∫
−
2
π
2
π
e
x
+
1
cos
(
x
)
7
d
x
.