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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2013 Oral Moscow Geometry Olympiad
2013 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
6
2
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minimal distance between circumcenters
Let
A
B
C
ABC
A
BC
be a triangle. On its sides
A
B
AB
A
B
and
B
C
BC
BC
are fixed points
C
1
C_1
C
1
and
A
1
A_1
A
1
, respectively. Find a point
P
P
P
on the circumscribed circle of triangle
A
B
C
ABC
A
BC
such that the distance between the centers of the circumscribed circles of the triangles
A
P
C
1
APC_1
A
P
C
1
and
C
P
A
1
CPA_1
CP
A
1
is minimal.
lines intersect on circle, isosceles trapezoid, incircles related
The trapezoid
A
B
C
D
ABCD
A
BC
D
is inscribed in the circle
w
w
w
(
A
D
/
/
B
C
AD // BC
A
D
//
BC
). The circles inscribed in the triangles
A
B
C
ABC
A
BC
and
A
B
D
ABD
A
B
D
touch the base of the trapezoid
B
C
BC
BC
and
A
D
AD
A
D
at points
P
P
P
and
Q
Q
Q
respectively. Points
X
X
X
and
Y
Y
Y
are the midpoints of the arcs
B
C
BC
BC
and
A
D
AD
A
D
of circle
w
w
w
that do not contain points
A
A
A
and
B
B
B
respectively. Prove that lines
X
P
XP
XP
and
Y
Q
YQ
Y
Q
intersect on the circle
w
w
w
.
5
2
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line MH passes through arc midpoint in a 60-45-75 triangle
In triangle
A
B
C
,
∠
C
=
6
0
o
,
∠
A
=
4
5
o
ABC, \angle C= 60^o, \angle A= 45^o
A
BC
,
∠
C
=
6
0
o
,
∠
A
=
4
5
o
. Let
M
M
M
be the midpoint of
B
C
,
H
BC, H
BC
,
H
be the orthocenter of triangle
A
B
C
ABC
A
BC
. Prove that line
M
H
MH
M
H
passes through the midpoint of arc
A
B
AB
A
B
of the circumcircle of triangle
A
B
C
ABC
A
BC
.
perpendicularity wanted, heights, midpoints, circumcenter related
In the acute-angled triangle
A
B
C
ABC
A
BC
, let
A
P
AP
A
P
and
B
Q
BQ
BQ
be the altitudes,
C
M
CM
CM
be the median . Point
R
R
R
is the midpoint of
C
M
CM
CM
. Line
P
Q
PQ
PQ
intersects line
A
B
AB
A
B
at
T
T
T
. Prove that
O
R
⊥
T
C
OR \perp TC
OR
⊥
TC
, where
O
O
O
is the center of the circumscribed circle of triangle
A
B
C
ABC
A
BC
.
4
2
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2 circumcircles intersect on angle bisector, equal segments in extensions
Let
A
B
C
ABC
A
BC
be a triangle. On the extensions of sides
A
B
AB
A
B
and
C
B
CB
CB
towards
B
B
B
, points
C
1
C_1
C
1
and
A
1
A_1
A
1
are taken, respectively, so that
A
C
=
A
1
C
=
A
C
1
AC = A_1C = AC_1
A
C
=
A
1
C
=
A
C
1
. Prove that circumscribed circles of triangles
A
B
A
1
ABA_1
A
B
A
1
and
C
B
C
1
CBC_1
CB
C
1
intersect on the bisector of angle
B
B
B
.
similar triangles on sides of a ABCD with perpendicular diagonals
Similar triangles
A
B
M
,
C
B
P
,
C
D
L
ABM, CBP, CDL
A
BM
,
CBP
,
C
D
L
and
A
D
K
ADK
A
DK
are built on the sides of the quadrilateral
A
B
C
D
ABCD
A
BC
D
with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that
P
K
=
M
L
PK = ML
P
K
=
M
L
.
3
2
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midpoint of KI lies on AC, angle bisectors related, right triangle
The bisectors
A
A
1
AA_1
A
A
1
and
C
C
1
CC_1
C
C
1
of the right triangle
A
B
C
ABC
A
BC
(
∠
B
=
9
0
o
\angle B = 90^o
∠
B
=
9
0
o
) intersect at point
I
I
I
. The line passing through the point
C
1
C_1
C
1
and perpendicular on the line
A
A
1
AA_1
A
A
1
intersects the line that passes through
A
1
A_1
A
1
and is perpendicular on
C
C
1
CC_1
C
C
1
, at the point
K
K
K
. Prove that the midpoint of the segment
K
I
KI
K
I
lies on segment
A
C
AC
A
C
.
is there polyhedron whose area ratio of any two faces is at least 2
Is there a polyhedron whose area ratio of any two faces is at least
2
2
2
?
2
2
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triangle split into 2 triangles with same sum of squares of sides
With a compass and a ruler, split a triangle into two smaller triangles with the same sum of squares of sides.
intersection point of internal common tangent lies on angle bisector
Inside the angle
A
O
D
AOD
A
O
D
, the rays
O
B
OB
OB
and
O
C
OC
OC
are drawn such that
∠
A
O
B
=
∠
C
O
D
.
\angle AOB = \angle COD.
∠
A
OB
=
∠
CO
D
.
Two circles are inscribed inside the angles
∠
A
O
B
\angle AOB
∠
A
OB
and
∠
C
O
D
\angle COD
∠
CO
D
. Prove that the intersection point of the common internal tangents of these circles lies on the bisector of the angle
A
O
D
AOD
A
O
D
.
1
2
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angle bisector perpendicular to the median, twice in the same figure
In triangle
A
B
C
ABC
A
BC
the angle bisector
A
K
AK
A
K
is perpendicular on the median is
C
L
CL
C
L
. Prove that in the triangle
B
K
L
BKL
B
K
L
also one of angle bisectors are perpendicular to one of the medians.
diagonal intersection of cyclic ABCD is incenter of BMC, circumcircles related
Diagonals of a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at point
O
O
O
. The circumscribed circles of triangles
A
O
B
AOB
A
OB
and
C
O
D
COD
CO
D
intersect at point
M
M
M
on the side
A
D
AD
A
D
. Prove that the point
O
O
O
is the center of the inscribed circle of the triangle
B
M
C
BMC
BMC
.