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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2010 Oral Moscow Geometry Olympiad
2010 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
6
2
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line touches a fixed circle., perpendicular bisectors and circumcircle related
Perpendicular bisectors of the sides
B
C
BC
BC
and
A
C
AC
A
C
of an acute-angled triangle
A
B
C
ABC
A
BC
intersect lines
A
C
AC
A
C
and
B
C
BC
BC
at points
M
M
M
and
N
N
N
. Let point
C
C
C
move along the circumscribed circle of triangle
A
B
C
ABC
A
BC
, remaining in the same half-plane relative to
A
B
AB
A
B
(while points
A
A
A
and
B
B
B
are fixed). Prove that line
M
N
MN
MN
touches a fixed circle.
concurrency, lines passing through midpoint of altitudes // OA_i
In a triangle
A
B
C
,
O
ABC, O
A
BC
,
O
is the center of the circumscribed circle. Line
a
a
a
passes through the midpoint of the altitude of the triangle from the vertex
A
A
A
and is parallel to
O
A
OA
O
A
. Similarly, the straight lines
b
b
b
and
c
c
c
are defined. Prove that these three lines intersect at one point.
5
2
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angle chasing inside a square ABCD, ML = KL given
Points
K
K
K
and
M
M
M
are taken on the sides
A
B
AB
A
B
and
C
D
CD
C
D
of square
A
B
C
D
ABCD
A
BC
D
, respectively, and on the diagonal
A
C
AC
A
C
- point
L
L
L
such that
M
L
=
K
L
ML = KL
M
L
=
K
L
. Let
P
P
P
be the intersection point of the segments
M
K
MK
M
K
and
B
D
BD
B
D
. Find the angle
∠
K
P
L
\angle KPL
∠
K
P
L
.
regular pyramid inscribed in a cylinder
All edges of a regular right pyramid are equal to
1
1
1
, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius
R
R
R
. Find all possible values of
R
R
R
.
4
2
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a right-angled triangle can be constructed from the segments AK, CM,MK
An isosceles triangle
A
B
C
ABC
A
BC
with base
A
C
AC
A
C
is given. Point
H
H
H
is the intersection of altitudes. On the sides
A
B
AB
A
B
and
B
C
BC
BC
, points
M
M
M
and
K
K
K
are selected, respectively, so that the angle
K
M
H
KMH
K
M
H
is right. Prove that a right-angled triangle can be constructed from the segments
A
K
,
C
M
AK, CM
A
K
,
CM
and
M
K
MK
M
K
.
perpendicular wanted, perpendiculars inside parallelogram
From the vertex
A
A
A
of the parallelogram
A
B
C
D
ABCD
A
BC
D
, the perpendiculars
A
M
,
A
N
AM,AN
A
M
,
A
N
on sides
B
C
,
C
D
BC,CD
BC
,
C
D
respectively.
P
P
P
is the intersection point of
B
N
BN
BN
and
D
M
DM
D
M
. Prove that the lines
A
P
AP
A
P
and
M
N
MN
MN
are perpendicular.
3
2
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collinear wanted, intersecting circles and tangents related
Two circles
w
1
w_1
w
1
and
w
2
w_2
w
2
intersect at points
A
A
A
and
B
B
B
. Tangents
ℓ
1
\ell_1
ℓ
1
and
ℓ
2
\ell_2
ℓ
2
respectively are drawn to them through point
A
A
A
. The perpendiculars dropped from point
B
B
B
to
ℓ
2
\ell_2
ℓ
2
and
ℓ
1
\ell_1
ℓ
1
intersects the circles
w
1
w_1
w
1
and
w
2
w_2
w
2
, respectively, at points
K
K
K
and
N
N
N
. Prove that points
K
,
A
K, A
K
,
A
and
N
N
N
lie on one straight line.
fixed point, S_{KMC} + S_{KAC}=S_{ABC}, area condition , two non constant points
On the sides
A
B
AB
A
B
and
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
, points
M
M
M
and
K
K
K
are taken, respectively, so that
S
K
M
C
+
S
K
A
C
=
S
A
B
C
S_{KMC} + S_{KAC}=S_{ABC}
S
K
MC
+
S
K
A
C
=
S
A
BC
. Prove that all such lines
M
K
MK
M
K
pass through one point.
2
2
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paperfolding a square paper of side 1 to measure a distance 5/6
Given a square sheet of paper with side
1
1
1
. Measure on this sheet a distance of
5
/
6
5/6
5/6
. (The sheet can be folded, including, along any segment with ends at the edges of the paper and unbend back, after unfolding, a trace of the fold line remains on the paper).
parallel lines by projections inside a cyclic ABCD
Quadrangle
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle. The perpendicular from the vertex
C
C
C
on the bisector of
∠
A
B
D
\angle ABD
∠
A
B
D
intersects the line
A
B
AB
A
B
at the point
C
1
C_1
C
1
. The perpendicular from the vertex
B
B
B
on the bisector of
∠
A
C
D
\angle ACD
∠
A
C
D
intersects the line
C
D
CD
C
D
at the point
B
1
B_1
B
1
. Prove that
B
1
C
1
∥
A
D
B_1C_1 \parallel AD
B
1
C
1
∥
A
D
.
1
2
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angle between AD and BE of equilaterals ABC and CDE
Two equilateral triangles
A
B
C
ABC
A
BC
and
C
D
E
CDE
C
D
E
have a common vertex (see fig). Find the angle between straight lines
A
D
AD
A
D
and
B
E
BE
BE
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convex cyclic n-gon cut into equal triangles by non-intersecting diagonals
Convex
n
n
n
-gon
P
P
P
, where
n
>
3
n> 3
n
>
3
, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of
n
n
n
if the
n
n
n
-gon is cyclic?