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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2003 Oral Moscow Geometry Olympiad
2003 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
1
1
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a triangle construction, angle, opposite side and median of other side
Construct a triangle given an angle, the side opposite the angle and the median to the other side (researching the number of solutions is not required).
6
1
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min no of lines, reflecting on a circle to cover each point in plane
A circle is located on the plane. What is the smallest number of lines you need to draw so that, symmetrically reflecting a given circle relative to these lines (in any order a finite number of times), it could cover any given point of the plane?
5
1
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concurrency wanted, centers of 3 rectangles related, wrt a triangle
Given triangle
A
B
C
ABC
A
BC
. Point
O
1
O_1
O
1
is the center of the
B
C
D
E
BCDE
BC
D
E
rectangle, constructed so that the side
D
E
DE
D
E
of the rectangle contains the vertex
A
A
A
of the triangle. Points
O
2
O_2
O
2
and
O
3
O_3
O
3
are the centers of rectangles constructed in the same way on the sides
A
C
AC
A
C
and
A
B
AB
A
B
, respectively. Prove that lines
A
O
1
,
B
O
2
AO_1, BO_2
A
O
1
,
B
O
2
and
C
O
3
CO_3
C
O
3
meet at one point.
4
1
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perpendicularity wanted, incenter, centroid, touchpoints with incircle, CA'= AB
In triangle
A
B
C
ABC
A
BC
,
M
M
M
is the point of intersection of the medians,
O
O
O
is the center of the inscribed circle,
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
are the touchpoints with the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, respectively. Prove that if
C
A
′
=
A
B
CA'= AB
C
A
′
=
A
B
, then
O
M
OM
OM
and
A
B
AB
A
B
are perpendicular.PS. There is a a typo
3
1
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concyclic wanted, <MBA=<LBC, BK=BC, BF=AB, circles with diameters AB,AC
Inside the segment
A
C
AC
A
C
, an arbitrary point
B
B
B
is selected and circles with diameters
A
B
AB
A
B
and
B
C
BC
BC
are constructed. Points
M
M
M
and
L
L
L
are chosen on the circles (in one half-plane with respect to
A
C
AC
A
C
), respectively, so that
∠
M
B
A
=
∠
L
B
C
\angle MBA = \angle LBC
∠
MB
A
=
∠
L
BC
. Points
K
K
K
and
F
F
F
are marked, respectively, on rays
B
M
BM
BM
and
B
L
BL
B
L
so that
B
K
=
B
C
BK = BC
B
K
=
BC
and
B
F
=
A
B
BF = AB
BF
=
A
B
. Prove that points
M
,
K
,
F
M, K, F
M
,
K
,
F
and
L
L
L
lie on the same circle.
2
1
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ratio chasing, <ABC = 90^o , <BAC =<CAD, AC = AD, DH altitude
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
,
∠
A
B
C
=
9
0
o
\angle ABC = 90^o
∠
A
BC
=
9
0
o
,
∠
B
A
C
=
∠
C
A
D
\angle BAC = \angle CAD
∠
B
A
C
=
∠
C
A
D
,
A
C
=
A
D
,
D
H
AC = AD, DH
A
C
=
A
D
,
DH
is the alltitude of the triangle
A
C
D
ACD
A
C
D
. In what ratio does the line
B
H
BH
B
H
divide the segment
C
D
CD
C
D
?