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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2014 Oral Moscow Geometry Olympiad
2014 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
6
2
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angle chasing candidate, inside a right and isosceles triangle
Inside an isosceles right triangle
A
B
C
ABC
A
BC
with hypotenuse
A
B
AB
A
B
a point
M
M
M
is taken such that the angle
∠
M
A
B
\angle MAB
∠
M
A
B
is
1
5
o
15 ^o
1
5
o
larger than the angle
∠
M
A
C
\angle MAC
∠
M
A
C
, and the angle
∠
M
C
B
\angle MCB
∠
MCB
is
1
5
o
15^o
1
5
o
larger than the angle
∠
M
B
C
\angle MBC
∠
MBC
. Find the angle
∠
B
M
C
\angle BMC
∠
BMC
.
2 lines connecting incenters with excenters concurrent with angle bisector
A convex quadrangle
A
B
C
D
ABCD
A
BC
D
is given. Let
I
I
I
and
J
J
J
be the circles of circles inscribed in the triangles
A
B
C
ABC
A
BC
and
A
D
C
ADC
A
D
C
, respectively, and
I
a
I_a
I
a
and
J
a
J_a
J
a
are the centers of the excircles circles of triangles
A
B
C
ABC
A
BC
and
A
D
C
ADC
A
D
C
, respectively (inscribed in the angles
B
A
C
BAC
B
A
C
and
D
A
C
DAC
D
A
C
, respectively). Prove that the intersection point
K
K
K
of the lines
I
J
a
IJ_a
I
J
a
and
J
I
a
JI_a
J
I
a
lies on the bisector of the angle
B
C
D
BCD
BC
D
.
5
2
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perimeter of the triangle DEF is two times larger than the side BC
Segment
A
D
AD
A
D
is the diameter of the circumscribed circle of an acute-angled triangle
A
B
C
ABC
A
BC
. Through the intersection of the altitudes of this triangle, a straight line was drawn parallel to the side
B
C
BC
BC
, which intersects sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
E
E
E
and
F
F
F
, respectively. Prove that the perimeter of the triangle
D
E
F
DEF
D
EF
is two times larger than the side
B
C
BC
BC
.
P in circumcircle of an equilateral ABC, AP cuts BC at A', area of A'B'C'
Given a regular triangle
A
B
C
ABC
A
BC
, whose area is
1
1
1
, and the point
P
P
P
on its circumscribed circle. Lines
A
P
,
B
P
,
C
P
AP, BP, CP
A
P
,
BP
,
CP
intersect, respectively, lines
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at points
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
. Find the area of the triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
.
4
2
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2 circumcircles concurrent with angle bisector
In triangle
A
B
C
ABC
A
BC
, the perpendicular bisectors of sides
A
B
AB
A
B
and
B
C
BC
BC
intersect side
A
C
AC
A
C
at points
P
P
P
and
Q
Q
Q
, respectively, with point
P
P
P
lying on the segment
A
Q
AQ
A
Q
. Prove that the circumscribed circles of the triangles
P
B
C
PBC
PBC
and
Q
B
A
QBA
QB
A
intersect on the bisector of the angle
P
B
Q
PBQ
PBQ
.
3 points collinear, lie on line perpendicular to Euler line
The medians
A
A
0
,
B
B
0
AA_0, BB_0
A
A
0
,
B
B
0
, and
C
C
0
CC_0
C
C
0
of the acute-angled triangle
A
B
C
ABC
A
BC
intersect at the point
M
M
M
, and heights
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
at point
H
H
H
. Tangent to the circumscribed circle of triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
at
C
1
C_1
C
1
intersects the line
A
0
B
0
A_0B_0
A
0
B
0
at the point
C
′
C'
C
′
. Points
A
′
A'
A
′
and
B
′
B'
B
′
are defined similarly. Prove that
A
′
,
B
′
A', B'
A
′
,
B
′
and
C
′
C'
C
′
lie on one line perpendicular to the line
M
H
MH
M
H
.
3
2
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convex pentagon in which each diagonal is equal to a side?
Is there a convex pentagon in which each diagonal is equal to a side?
2 lines concurrent with circumcircle, angle bisectors, circumcircles related
The bisectors
A
A
1
AA_1
A
A
1
and
C
C
1
CC_1
C
C
1
of triangle
A
B
C
ABC
A
BC
intersect at point
I
I
I
. The circumscribed circles of triangles
A
I
C
1
AIC_1
A
I
C
1
and
C
I
A
1
CIA_1
C
I
A
1
intersect the arcs
A
C
AC
A
C
and
B
C
BC
BC
(not containing points
B
B
B
and
A
A
A
respectively) of the circumscribed circle of triangle
A
B
C
ABC
A
BC
at points
C
2
C_2
C
2
and
A
2
A_2
A
2
, respectively. Prove that lines
A
1
A
2
A_1A_2
A
1
A
2
and
C
1
C
2
C_1C_2
C
1
C
2
intersect on the circumscribed circle of triangle
A
B
C
ABC
A
BC
.
2
2
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in a parallelogram, equal segments given and wanted
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram. On side
A
B
AB
A
B
, point
M
M
M
is taken so that
A
D
=
D
M
AD = DM
A
D
=
D
M
. On side
A
D
AD
A
D
point
N
N
N
is taken so that
A
B
=
B
N
AB = BN
A
B
=
BN
. Prove that
C
M
=
C
N
CM = CN
CM
=
CN
.
Is it possible to cut a regular triangular prism into two equal pyramids?
Is it possible to cut a regular triangular prism into two equal pyramids?
1
2
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circumcenter coincides with excenter , <A=45^o
In triangle
A
B
C
,
∠
A
=
4
5
o
,
B
H
ABC, \angle A= 45^o, BH
A
BC
,
∠
A
=
4
5
o
,
B
H
is the altitude, the point
K
K
K
lies on the
A
C
AC
A
C
side, and
B
C
=
C
K
BC = CK
BC
=
C
K
. Prove that the center of the circumscribed circle of triangle
A
B
K
ABK
A
B
K
coincides with the center of an excircle of triangle
B
C
H
BCH
BC
H
.
concurrency in a trapezoid, midpoints, height related
In trapezoid
A
B
C
D
ABCD
A
BC
D
:
B
C
<
A
D
,
A
B
=
C
D
,
K
BC <AD, AB = CD, K
BC
<
A
D
,
A
B
=
C
D
,
K
is midpoint of
A
D
,
M
AD, M
A
D
,
M
is midpoint of
C
D
,
C
H
CD, CH
C
D
,
C
H
is height. Prove that lines
A
M
,
C
K
AM, CK
A
M
,
C
K
and
B
H
BH
B
H
intersect at one point.