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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2019 Oral Moscow Geometry Olympiad
2019 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
6
2
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perpendicularity related to excircles
In the acute triangle
A
B
C
ABC
A
BC
, the point
I
c
I_c
I
c
is the center of excircle on the side
A
B
AB
A
B
,
A
1
A_1
A
1
and
B
1
B_1
B
1
are the tangency points of the other two excircles with sides
B
C
BC
BC
and
C
A
CA
C
A
, respectively,
C
′
C'
C
′
is the point on the circumcircle diametrically opposite to point
C
C
C
. Prove that the lines
I
c
C
′
I_cC'
I
c
C
′
and
A
1
B
1
A_1B_1
A
1
B
1
are perpendicular.
cosines of the flat angles of the trihedral angle is −1, sum of dihedral ?
The sum of the cosines of the flat angles of the trihedral angle is
−
1
-1
−
1
. Find the sum of its dihedral angles.
5
2
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locus of intersections of perpendicular bisectors of two equal segments is line
Given the segment
P
Q
PQ
PQ
and a circle . A chord
A
B
AB
A
B
moves around the circle, equal to
P
Q
PQ
PQ
. Let
T
T
T
be the intersection point of the perpendicular bisectors of the segments
A
P
AP
A
P
and
B
Q
BQ
BQ
. Prove that all points of
T
T
T
thus obtained lie on one line.
two lines concurrent with a circumcircle
On sides
A
B
AB
A
B
and
B
C
BC
BC
of a non-isosceles triangle
A
B
C
ABC
A
BC
are selected points
C
1
C_1
C
1
and
A
1
A_1
A
1
such that the quadrilateral
A
C
1
A
1
C
AC_1A_1C
A
C
1
A
1
C
is cyclic. Lines
C
C
1
CC_1
C
C
1
and
A
A
1
AA_1
A
A
1
intersect at point
P
P
P
. Line
B
P
BP
BP
intersects the circumscribed circle of triangle
A
B
C
ABC
A
BC
at the point
Q
Q
Q
. Prove that the lines
Q
C
1
QC_1
Q
C
1
and
C
M
CM
CM
, where
M
M
M
is the midpoint of
A
1
C
1
A_1C_1
A
1
C
1
, intersect at the circumscribed circles of triangle
A
B
C
ABC
A
BC
.
4
2
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tangent circumcircles
The perpendicular bisector of the bisector
B
L
BL
B
L
of the triangle
A
B
C
ABC
A
BC
intersects the bisectors of its external angles
A
A
A
and
C
C
C
at points
P
P
P
and
Q
Q
Q
, respectively. Prove that the circle circumscribed around the triangle
P
B
Q
PBQ
PBQ
is tangent to the circle circumscribed around the triangle
A
B
C
ABC
A
BC
.
angle between touchpoints of excircles of a right triangle
Given a right triangle
A
B
C
ABC
A
BC
(
∠
C
=
9
0
o
\angle C=90^o
∠
C
=
9
0
o
). The
C
C
C
-excircle touches the hypotenuse
A
B
AB
A
B
at point
C
1
,
A
1
C_1, A_1
C
1
,
A
1
is the touchpoint of
B
B
B
-excircle with line
B
C
,
B
1
BC, B_1
BC
,
B
1
is the touchpoint of
A
A
A
-excircle with line
A
C
AC
A
C
. Find the angle
∠
A
1
C
1
B
1
\angle A_1C_1B_1
∠
A
1
C
1
B
1
.
3
2
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<B=60^o, circumcenter, orthocenter, concyclic wanted
In the acute triangle
A
B
C
,
∠
A
B
C
=
6
0
o
,
O
ABC, \angle ABC = 60^o , O
A
BC
,
∠
A
BC
=
6
0
o
,
O
is the center of the circumscribed circle and
H
H
H
is the orthocenter. The angle bisector
B
L
BL
B
L
intersects the circumscribed circle at the point
W
,
X
W, X
W
,
X
is the intersection point of segments
W
H
WH
W
H
and
A
C
AC
A
C
. Prove that points
O
,
L
,
X
O, L, X
O
,
L
,
X
and
H
H
H
lie on the same circle.
triangle construction, given circumcircle of BHC, A, base of B altitude
Restore the acute triangle
A
B
C
ABC
A
BC
given the vertex
A
A
A
, the foot of the altitude drawn from the vertex
B
B
B
and the center of the circle circumscribed around triangle
B
H
C
BHC
B
H
C
(point
H
H
H
is the orthocenter of triangle
A
B
C
ABC
A
BC
).
2
2
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parallelogram on side of a triangle, concurrent lines wanted
On the side
A
C
AC
A
C
of the triangle
A
B
C
ABC
A
BC
in the external side is constructed the parallelogram
A
C
D
E
ACDE
A
C
D
E
. Let
O
O
O
be the intersection point of its diagonals,
N
N
N
and
K
K
K
be midpoints of BC and BA respectively. Prove that lines
D
K
,
E
N
DK, EN
DK
,
EN
and
B
O
BO
BO
intersect at one point.
quadrilateral with same angles, equal diagonals, are equal?
The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?
1
2
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distance of a point from incircle equals to a diameter of incircle
In the triangle
A
B
C
,
I
ABC, I
A
BC
,
I
is the center of the inscribed circle, point
M
M
M
lies on the side of
B
C
BC
BC
, with
∠
B
I
M
=
9
0
o
\angle BIM = 90^o
∠
B
I
M
=
9
0
o
. Prove that the distance from point
M
M
M
to line
A
B
AB
A
B
is equal to the diameter of the circle inscribed in triangle
A
B
C
ABC
A
BC
angle chasing related to circle inscribed in square
Circle inscribed in square
A
B
C
D
ABCD
A
BC
D
, is tangent to sides
A
B
AB
A
B
and
C
D
CD
C
D
at points
M
M
M
and
K
K
K
respectively. Line
B
K
BK
B
K
intersects this circle at the point
L
,
X
L, X
L
,
X
is the midpoint of
K
L
KL
K
L
. Find the angle
∠
M
X
K
\angle MXK
∠
MX
K
.