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Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2023 Oral Moscow Geometry Olympiad
2023 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
4
2
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The center of (IPQ) is on BC
Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
, tangent to sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
E
E
E
and
F
F
F
, respectively. The lines through
E
E
E
and
F
F
F
parallel to
A
I
AI
A
I
intersect lines
B
I
BI
B
I
and
C
I
CI
C
I
at points
P
P
P
and
Q
Q
Q
, respectively. Prove that the center of the circumcircle of triangle
I
P
Q
IPQ
I
PQ
lies on line
B
C
BC
BC
.
concurrency in 3D, lines connecting tyoucpoints of incircle and excircle
Given isosceles tetrahedron
P
A
B
C
PABC
P
A
BC
(faces are equal triangles). Let
A
0
A_0
A
0
,
B
0
B_0
B
0
and
C
0
C_0
C
0
be the touchpoints of the circle inscribed in the triangle
A
B
C
ABC
A
BC
with sides
B
C
BC
BC
,
A
C
AC
A
C
and
A
B
AB
A
B
respectively,
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
are the touchpoints of the excircles of triangles
P
C
A
PCA
PC
A
,
P
A
B
PAB
P
A
B
and
P
B
C
PBC
PBC
with extensions of sides
P
A
PA
P
A
,
P
B
PB
PB
and
P
C
PC
PC
, respectively (beyond points
A
A
A
,
B
B
B
,
C
C
C
). Prove that the lines
A
0
A
1
A_0A_1
A
0
A
1
,
B
0
B
1
B_0B_1
B
0
B
1
and
C
0
C
1
C_0C_1
C
0
C
1
intersect at one point.
1
2
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Trapezoid and bisection
In trapezoid
A
B
C
D
ABCD
A
BC
D
with bases
A
D
,
B
C
AD, BC
A
D
,
BC
,
A
D
=
2
B
C
AD = 2BC
A
D
=
2
BC
and
M
M
M
is midpoint of
A
B
AB
A
B
. Prove that line
B
D
BD
B
D
passes through the midpoint of segment
C
M
CM
CM
.
Geo with angle 60 degrees
In triangle ABC
∠
A
B
C
=
6
0
o
\angle ABC=60^{o}
∠
A
BC
=
6
0
o
and
O
O
O
is the center of the circumscribed circle. The bisector
B
L
BL
B
L
intersects the circumscribed circle at the point
W
W
W
. Prove that
O
W
OW
O
W
is tangent to
(
B
O
L
)
(BOL)
(
BO
L
)
2
2
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rectangle with ratio of sides \sqrt2 by square sheet of paper
There is a square sheet of paper. How to get a rectangular sheet of paper with an aspect ratio equal to
2
\sqrt2
2
? (There are no tools, the sheet can only be bent.)
Find the locus
Points
X
1
X_1
X
1
and
X
2
X_2
X
2
move along fixed circles with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively, so that
O
1
X
1
∥
O
2
X
2
O_1X_1 \parallel O_2X_2
O
1
X
1
∥
O
2
X
2
. Find the locus of the intersection point of lines
O
1
X
2
O_1X_2
O
1
X
2
and
O
2
X
1
O_2X_1
O
2
X
1
.
3
2
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Segment is bisected by the NPC
Given is a triangle
A
B
C
ABC
A
BC
and
M
M
M
is the midpoint of the minor arc
B
C
BC
BC
. Let
M
1
M_1
M
1
be the reflection of
M
M
M
with respect to side
B
C
BC
BC
. Prove that the nine-point circle bisects
A
M
1
AM_1
A
M
1
.
Conditional geo
In an acute triangle
A
B
C
ABC
A
BC
the line
O
I
OI
O
I
is parallel to side
B
C
BC
BC
. Prove that the center of the nine-point circle of triangle
A
B
C
ABC
A
BC
lies on the line
M
I
MI
M
I
, where
M
M
M
is the midpoint of
B
C
BC
BC
.
5
2
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Another geo with angle 60
Altitudes
B
B
1
BB_1
B
B
1
and
C
C
1
CC_1
C
C
1
of acute triangle
A
B
C
ABC
A
BC
intersect at
H
H
H
, and
∠
A
=
6
0
o
\angle A = 60^{o}
∠
A
=
6
0
o
,
A
B
<
A
C
AB < AC
A
B
<
A
C
. The median
A
M
AM
A
M
intersects the circumcircle of
A
B
C
ABC
A
BC
at point
K
K
K
;
L
L
L
is the midpoint of the arc
B
C
BC
BC
of the circumcircle that does not contain point
A
A
A
; lines
B
1
C
1
B_1C_1
B
1
C
1
and
B
C
BC
BC
intersect at point
E
E
E
. Prove that
∠
E
H
L
=
∠
A
B
K
\angle EHL = \angle ABK
∠
E
H
L
=
∠
A
B
K
.
Two tangent circles in an orthocenter configuration
In an acute-angled triangle
A
B
C
ABC
A
BC
with orthocenter
H
H
H
, the line
A
H
AH
A
H
cuts
B
C
BC
BC
at point
A
1
A_1
A
1
. Let
Γ
\Gamma
Γ
be a circle centered on side
A
B
AB
A
B
tangent to
A
A
1
AA_1
A
A
1
at point
H
H
H
. Prove that
Γ
\Gamma
Γ
is tangent to the circumscribed circle of triangle
A
M
A
1
AMA_1
A
M
A
1
, where
M
M
M
is the midpoint of
A
C
AC
A
C
.
6
2
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Fixed point geo
Given a circle
Ω
\Omega
Ω
tangent to side
A
B
AB
A
B
of angle
∠
B
A
C
\angle BAC
∠
B
A
C
and lying outside this angle. We consider circles
w
w
w
inscribed in angle
B
A
C
BAC
B
A
C
. The internal tangent of
Ω
\Omega
Ω
and
w
w
w
, different from
A
B
AB
A
B
, touches
w
w
w
at a point
K
K
K
. Let
L
L
L
be the point of tangency of
w
w
w
and
A
C
AC
A
C
. Prove that all such lines
K
L
KL
K
L
pass through a fixed point without depending on the choice of circle
w
w
w
.
Many angle conditions
Points
C
1
C_1
C
1
and
C
2
C_2
C
2
lie on side
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
, where the point
C
1
C_1
C
1
belongs to the segment
A
C
2
AC_2
A
C
2
and
∠
A
C
C
1
=
∠
B
C
C
2
\angle ACC_1= \angle BCC_2
∠
A
C
C
1
=
∠
BC
C
2
. On segments
C
C
1
CC_1
C
C
1
and
C
C
2
CC_2
C
C
2
points
A
′
A'
A
′
and
B
′
B'
B
′
are taken such that
∠
C
A
A
′
=
∠
C
B
B
′
=
∠
C
1
C
C
2
\angle CAA'= \angle CBB' = \angle C_1CC_2
∠
C
A
A
′
=
∠
CB
B
′
=
∠
C
1
C
C
2
. Prove that the center of the circle
(
C
A
′
B
′
)
(CA'B')
(
C
A
′
B
′
)
lies on the perpendicular bisector of the segment
A
B
AB
A
B
.