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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2011 Oral Moscow Geometry Olympiad
2011 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
4
2
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perpendicular passes through midpoint, in trapezoid ABCD, AB = BC = CD
In the trapezoid
A
B
C
D
,
A
B
=
B
C
=
C
D
,
C
H
ABCD, AB = BC = CD, CH
A
BC
D
,
A
B
=
BC
=
C
D
,
C
H
is the altitude. Prove that the perpendicular from
H
H
H
on
A
C
AC
A
C
passes through the midpoint of
B
D
BD
B
D
.
rigid flat triangle T with area <4, may be inserted in triangular hole of area 3
Prove that any rigid flat triangle
T
T
T
of area less than
4
4
4
can be inserted through a triangular hole
Q
Q
Q
with area
3
3
3
.
6
2
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triangle inscribed in triangle, 1 of 2 smallest sides is the inner's triangle
One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.
concurrency wanted, 2 circumcircles and tangents related
Let
A
A
1
,
B
B
1
AA_1 , BB_1
A
A
1
,
B
B
1
, and
C
C
1
CC_1
C
C
1
be the altitudes of the non-isosceles acute-angled triangle
A
B
C
ABC
A
BC
. The circles circumscibred around the triangles
A
B
C
ABC
A
BC
and
A
1
B
1
C
A_1 B_1 C
A
1
B
1
C
intersect again at the point
P
,
Z
P , Z
P
,
Z
is the intersection point of the tangents to the circumscribed circle of the triangle
A
B
C
ABC
A
BC
conducted at points
A
A
A
and
B
B
B
. Prove that lines
A
P
,
B
C
AP , BC
A
P
,
BC
and
Z
C
1
ZC_1
Z
C
1
are concurrent.
5
2
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<BDC when AC\perp BD, <BCA = 10^o,<BDA = 20^o, <BAC = 40^o
In a convex quadrilateral
A
B
C
D
,
A
C
⊥
B
D
,
∠
B
C
A
=
1
0
o
,
∠
B
D
A
=
2
0
o
,
∠
B
A
C
=
4
0
o
ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o
A
BC
D
,
A
C
⊥
B
D
,
∠
BC
A
=
1
0
o
,
∠
B
D
A
=
2
0
o
,
∠
B
A
C
=
4
0
o
. Find
∠
B
D
C
\angle BDC
∠
B
D
C
.
collinear wanted, circumcircles related to an isosceles
Let
A
A
1
AA _1
A
A
1
and
B
B
1
BB_1
B
B
1
be the altitudes of an isosceles acute-angled triangle
A
B
C
,
M
ABC, M
A
BC
,
M
the midpoint of
A
B
AB
A
B
. The circles circumscribed around the triangles
A
M
A
1
AMA_1
A
M
A
1
and
B
M
B
1
BMB_1
BM
B
1
intersect the lines
A
C
AC
A
C
and
B
C
BC
BC
at points
K
K
K
and
L
L
L
, respectively. Prove that
K
,
M
K, M
K
,
M
, and
L
L
L
lie on the same line.
3
2
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dividing a diagonal of a 2x2 square into 6 equal parts with unmarked ruler
A
2
×
2
2\times 2
2
×
2
square was cut from a squared sheet of paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into
6
6
6
equal parts.
concurrency related to a trapezoid and a circle
A non-isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
(
A
B
/
/
C
D
AB // CD
A
B
//
C
D
) is given. An arbitrary circle passing through points
A
A
A
and
B
B
B
intersects the sides of the trapezoid at points
P
P
P
and
Q
Q
Q
, and the intersect the diagonals at points
M
M
M
and
N
N
N
. Prove that the lines
P
Q
,
M
N
PQ, MN
PQ
,
MN
and
C
D
CD
C
D
are concurrent.
2
2
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angle chasing inside an isosceles triangle, equal segments and right angle
In an isosceles triangle
A
B
C
ABC
A
BC
(
A
B
=
A
C
AB=AC
A
B
=
A
C
) on the side
B
C
BC
BC
, point
M
M
M
is marked so that the segment
C
M
CM
CM
is equal to the altitude of the triangle drawn on this side, and on the side
A
B
AB
A
B
, point
K
K
K
is marked so that the angle
∠
K
M
C
\angle KMC
∠
K
MC
is right. Find the angle
∠
A
C
K
\angle ACK
∠
A
C
K
.
2011 lines equidistant from line and not intersecting it, plane related
Line
ℓ
\ell
ℓ
intersects the plane
a
a
a
. It is known that in this plane there are
2011
2011
2011
straight lines equidistant from
ℓ
\ell
ℓ
and not intersecting
ℓ
\ell
ℓ
. Is it true that
ℓ
\ell
ℓ
is perpendicular to
a
a
a
?
1
2
Hide problems
MK is equal and perpendicular to the diagonal of the rectangle ABCD
The bisector of angle
B
B
B
and the bisector of external angle
D
D
D
of rectangle
A
B
C
D
ABCD
A
BC
D
intersect side
A
D
AD
A
D
and line
A
B
AB
A
B
at points
M
M
M
and
K
K
K
, respectively. Prove that the segment
M
K
MK
M
K
is equal and perpendicular to the diagonal of the rectangle.
AB is tangent to the cicrumcircle of DEC'
A
D
AD
A
D
and
B
E
BE
BE
are the altitudes of the triangle
A
B
C
ABC
A
BC
. It turned out that the point
C
′
C'
C
′
, symmetric to the vertex
C
C
C
wrt to the midpoint of the segment
D
E
DE
D
E
, lies on the side
A
B
AB
A
B
. Prove that
A
B
AB
A
B
is tangent to the circle circumscribed around the triangle
D
E
C
′
DEC'
D
E
C
′
.