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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2012 Oral Moscow Geometry Olympiad
2012 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
6
2
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triangle construction, given orthocenter, cuts of median, angle-bisector
Restore the triangle with a compass and a ruler given the intersection point of altitudes and the feet of the median and angle bisectors drawn to one side. (No research required.)
tangent circumcircles wanted
Tangents drawn to the circumscribed circle of an acute-angled triangle
A
B
C
ABC
A
BC
at points
A
A
A
and
C
C
C
, intersect at point
Z
Z
Z
. Let
A
A
1
,
C
C
1
AA_1, CC_1
A
A
1
,
C
C
1
be altitudes. Line
A
1
C
1
A_1C_1
A
1
C
1
intersects
Z
A
,
Z
C
ZA, ZC
Z
A
,
ZC
at points
X
X
X
and
Y
Y
Y
, respectively. Prove that the circumscribed circles of the triangles
A
B
C
ABC
A
BC
and
X
Y
Z
XYZ
X
Y
Z
are tangent.
5
2
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fixed point, related to circumcircles and orthocenter
Given a circle and a chord
A
B
AB
A
B
, different from the diameter. Point
C
C
C
moves along the large arc
A
B
AB
A
B
. The circle passing through passing through points
A
,
C
A, C
A
,
C
and point
H
H
H
of intersection of altitudes of of the triangle
A
B
C
ABC
A
BC
, re-intersects the line
B
C
BC
BC
at point
P
P
P
. Prove that line
P
H
PH
P
H
passes through a fixed point independent of the position of point
C
C
C
.
min sum of distances of a point of circle from equidistant points from center
Inside the circle with center
O
O
O
, points
A
A
A
and
B
B
B
are marked so that
O
A
=
O
B
OA = OB
O
A
=
OB
. Draw a point
M
M
M
on the circle from which the sum of the distances to points
A
A
A
and
B
B
B
is the smallest among all possible.
4
2
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OI \perp AC, where I incenter and O circumcenter of excenters I_A,I_C, and I
In triangle
A
B
C
ABC
A
BC
, point
I
I
I
is the center of the inscribed circle points, points
I
A
I_A
I
A
and
I
C
I_C
I
C
are the centers of the excircles, tangent to sides
B
C
BC
BC
and
A
B
AB
A
B
, respectively. Point
O
O
O
is the center of the circumscribed circle of triangle
I
I
A
I
C
II_AI_C
I
I
A
I
C
. Prove that
O
I
⊥
A
C
OI \perp AC
O
I
⊥
A
C
lines forming largeast angles with polyhedron faces and concurrent lines
Inside the convex polyhedron, the point
P
P
P
and several lines
ℓ
1
,
ℓ
2
,
.
.
.
,
ℓ
n
\ell_1,\ell_2, ..., \ell_n
ℓ
1
,
ℓ
2
,
...
,
ℓ
n
passing through
P
P
P
and not lying in the same plane. To each face of the polyhedron we associate one of the lines
l
1
,
l
2
,
.
.
.
,
l
n
l_1, l_2, ..., l_n
l
1
,
l
2
,
...
,
l
n
that forms the largest angle with the plane of this face (if there are there are several direct ones, we will choose any of them). Prove that there is a face that intersects with its corresponding line.
3
2
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line of isotomic of incircle touchpoints in equilateral is tangent to incircle
Given an equilateral triangle
A
B
C
ABC
A
BC
and a straight line
ℓ
\ell
ℓ
, passing through its center. Intersection points of this line with sides
A
B
AB
A
B
and
B
C
BC
BC
are reflected wrt to the midpoints of these sides respectively. Prove that the line passing through the resulting points, touches the inscribed circle triangle
A
B
C
ABC
A
BC
.
orthocenter of the triangle DEH lies on segment CP
H
H
H
is the intersection point of the heights
A
A
′
AA'
A
A
′
and
B
B
′
BB'
B
B
′
of the acute-angled triangle
A
B
C
ABC
A
BC
. A straight line, perpendicular to
A
B
AB
A
B
, intersects these heights at points
D
D
D
and
E
E
E
, and side
A
B
AB
A
B
at point
P
P
P
. Prove that the orthocenter of the triangle
D
E
H
DEH
D
E
H
lies on segment
C
P
CP
CP
.
2
2
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equal polygons with vertices of one interior or on border of other, coincide?
Two equal polygons
F
F
F
and
F
′
F'
F
′
are given on the plane. It is known that the vertices of the polygon
F
F
F
belong to
F
′
F'
F
′
(may lie inside it or on the border). Is it true that all the vertices of these polygons coincide?
<A = <C = 90^o, AB = AE, BC = CD, AC = 1, area of convex ABCD
In the convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
:
∠
A
=
∠
C
=
9
0
o
\angle A = \angle C = 90^o
∠
A
=
∠
C
=
9
0
o
,
A
B
=
A
E
,
B
C
=
C
D
,
A
C
=
1
AB = AE, BC = CD, AC = 1
A
B
=
A
E
,
BC
=
C
D
,
A
C
=
1
. Find the area of the pentagon.
1
2
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trapezoid AD//BC, AB = BC = BD, altitude BK cuts AC at M, <CDM?
In trapezoid
A
B
C
D
ABCD
A
BC
D
, the sides
A
D
AD
A
D
and
B
C
BC
BC
are parallel, and
A
B
=
B
C
=
B
D
AB = BC = BD
A
B
=
BC
=
B
D
. The height
B
K
BK
B
K
intersects the diagonal
A
C
AC
A
C
at
M
M
M
. Find
∠
C
D
M
\angle CDM
∠
C
D
M
.
incenter inside the midpoint's line triangle?
Is it true that the center of the inscribed circle of the triangle lies inside the triangle formed by the lines of connecting it's midpoints?