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Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2021 Oral Moscow Geometry Olympiad
2021 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
2
2
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midline of trapezoid construction, only ruler, one base is twice the other
A trapezoid is given in which one base is twice as large as the other. Use one ruler (no divisions) to draw the midline of this trapezoid.
criterion for congurent quadrilaterals or not?
Two quadrangles have equal areas, perimeters and corresponding angles. Are such quadrilaterals necessarily congurent ?
1
2
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tan<ABD wanted on a grid paper
Points
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
have been marked on checkered paper (see fig.). Find the tangent of the angle
A
B
D
ABD
A
B
D
. https://cdn.artofproblemsolving.com/attachments/6/1/eeb98ccdee801361f9f66b8f6b2da4714e659f.png
ad= bc, distances from a point of circumcircle to lines of inscribed ABCD
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle,
E
E
E
is an arbitrary point of this circle. It is known that distances from point
E
E
E
to lines
A
B
,
A
C
,
B
D
AB, AC, BD
A
B
,
A
C
,
B
D
and
C
D
CD
C
D
are equal to
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
respectively. Prove that
a
d
=
b
c
ad= bc
a
d
=
b
c
.
6
2
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MH intersects the angle bisector
Point
M
M
M
is a midpoint of side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
and
H
H
H
is the orthocenter of
A
B
C
ABC
A
BC
.
M
H
MH
M
H
intersects the
A
A
A
-angle bisector at
Q
Q
Q
. Points
X
X
X
and
Y
Y
Y
are the projections of
Q
Q
Q
on sides
A
B
AB
A
B
and
A
C
AC
A
C
. Prove that
X
Y
XY
X
Y
passes through
H
H
H
.
An equilateral triangle inscribed in a square
A
B
C
D
ABCD
A
BC
D
is a square and
X
Y
Z
XYZ
X
Y
Z
is an equilateral triangle such that
X
X
X
lies on
A
B
AB
A
B
,
Y
Y
Y
lies on
B
C
BC
BC
and
Z
Z
Z
lies on
D
A
DA
D
A
. Line through the centers of
A
B
C
D
ABCD
A
BC
D
and
X
Y
Z
XYZ
X
Y
Z
intersects
C
D
CD
C
D
at
T
T
T
. Find angle
C
T
Y
CTY
CT
Y
4
2
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three lines throught vertices and circumcenters are concurrent
On the diagonal
A
C
AC
A
C
of cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
a point
E
E
E
is chosen such that
∠
A
B
E
=
∠
C
B
D
\angle ABE = \angle CBD
∠
A
BE
=
∠
CB
D
. Points
O
,
O
1
,
O
2
O,O_1,O_2
O
,
O
1
,
O
2
are the circumcircles of triangles
A
B
C
,
A
B
E
ABC, ABE
A
BC
,
A
BE
and
C
B
E
CBE
CBE
respectively. Prove that lines
D
O
,
A
O
1
DO,AO_{1}
D
O
,
A
O
1
and
C
O
2
CO_{2}
C
O
2
are concurrent.
Sphere touches all edges of an octahedron STABCD. Prove that ABCD are complanar
Points
S
T
A
B
C
D
STABCD
ST
A
BC
D
in space form a convex octahedron with faces
S
A
B
,
S
B
C
,
S
C
D
,
S
D
A
,
T
A
B
,
T
B
C
,
T
C
D
,
T
D
A
SAB,SBC,SCD,SDA,TAB,TBC,TCD,TDA
S
A
B
,
SBC
,
SC
D
,
S
D
A
,
T
A
B
,
TBC
,
TC
D
,
T
D
A
such that there exists a sphere that is tangent to all of its edges. Prove that
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
lie in one plane.
3
2
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Quadrilateral with two equal angles, perpendicularity wanted
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral such that
∠
A
=
∠
C
<
9
0
∘
\angle A = \angle C < 90^{\circ}
∠
A
=
∠
C
<
9
0
∘
and
∠
A
B
D
=
9
0
∘
\angle ABD = 90^{\circ}
∠
A
B
D
=
9
0
∘
.
M
M
M
is the midpoint of
A
C
AC
A
C
. Prove that
M
B
MB
MB
is perpendicular to
C
D
CD
C
D
.
A moves on major arc BC and some point is on the same distance from BC
Circle
(
O
)
(O)
(
O
)
and its chord
B
C
BC
BC
are given. Point
A
A
A
moves on the major arc
B
C
BC
BC
.
A
L
AL
A
L
is the angle bisector in a triangle
A
B
C
ABC
A
BC
. Show that the disctance from the circumcenter of triangle
A
O
L
AOL
A
O
L
to the line
B
C
BC
BC
does not depend on the position of point
A
A
A
.
5
2
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geo inequality with inscribed trapezoid
The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.
Two triangles have the same nine point centers
Let
A
B
C
ABC
A
BC
be a triangle,
I
I
I
and
O
O
O
be its incenter and circumcenter respectively.
A
′
A'
A
′
is symmetric to
O
O
O
with respect to line
A
I
AI
A
I
. Points
B
′
B'
B
′
and
C
′
C'
C
′
are defined similarly. Prove that the nine-point centers of triangles
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
coincide.