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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2007 Oral Moscow Geometry Olympiad
2007 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
6
2
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fixed circle, all segments XY are tangent to same circle, projections
A point
P
P
P
is fixed inside the circle.
C
C
C
is an arbitrary point of the circle,
A
B
AB
A
B
is a chord passing through point
B
B
B
and perpendicular to the segment
B
C
BC
BC
. Points
X
X
X
and
Y
Y
Y
are projections of point
B
B
B
onto lines
A
C
AC
A
C
and
B
C
BC
BC
. Prove that all line segments
X
Y
XY
X
Y
are tangent to the same circle.(A. Zaslavsky)
fixed point wanted, intersection of tangents to circle, perpendicular rays
A circle and a point
P
P
P
inside it are given. Two arbitrary perpendicular rays starting at point
P
P
P
intersect the circle at points
A
A
A
and
B
B
B
. Point
X
X
X
is the projection of point
P
P
P
onto line
A
B
,
Y
AB, Y
A
B
,
Y
is the intersection point of tangents to the circle drawn through points
A
A
A
and
B
B
B
. Prove that all lines
X
Y
XY
X
Y
pass through the same point.(A. Zaslavsky)
5
2
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3 lines parallel wanted, symmetric of vertices wrt opposite sides
Given triangle
A
B
C
ABC
A
BC
. Points
A
1
,
B
1
A_1,B_1
A
1
,
B
1
and
C
1
C_1
C
1
are symmetric to its vertices with respect to opposite sides.
C
2
C_2
C
2
is the intersection point of lines
A
B
1
AB_1
A
B
1
and
B
A
1
BA_1
B
A
1
. Points
A
2
A_2
A
2
and
B
2
B_2
B
2
are defined similarly. Prove that the lines
A
1
A
2
,
B
1
B
2
A_1 A_2, B_1 B_2
A
1
A
2
,
B
1
B
2
and
C
1
C
2
C_1 C_2
C
1
C
2
are parallel.(A. Zaslavsky)
projections of point P onto lateral faces of pyramid are concyclic
At the base of the quadrangular pyramid
S
A
B
C
D
SABCD
S
A
BC
D
lies the quadrangle
A
B
C
D
ABCD
A
BC
D
. whose diagonals are perpendicular and intersect at point
P
P
P
, and
S
P
SP
SP
is the altitude of the pyramid. Prove that the projections of the point
P
P
P
onto the lateral faces of the pyramid lie on the same circle.(A. Zaslavsky)
4
2
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EF tangent to circumcircle of ABC, circumcircle of BIC related, incenter I
Let
I
I
I
be the center of a circle inscribed in triangle
A
B
C
ABC
A
BC
. The circle circumscribed about the triangle
B
I
C
BIC
B
I
C
intersects lines
A
B
AB
A
B
and
A
C
AC
A
C
at points
E
E
E
and
F
F
F
, respectively. Prove that the line
E
F
EF
EF
touches the circle inscribed in the triangle
A
B
C
ABC
A
BC
.
equal segments connecting opposite sides of hexagon wanted, equilateral
The midpoints of the opposite sides of the hexagon are connected by segments. It turned out that the points of pairwise intersection of these segments form an equilateral triangle. Prove that the drawn segments are equal.(M. Volchkevich)
3
2
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isosceles trapezoid criterion, sum of lengths of sides and diagonals related
In a trapezoid, the sum of the lengths of the side and the diagonal is equal to the sum of the lengths of the other side and the other diagonal. Prove that the trapezoid is isosceles.
paralellogram construction, 3 midpoints given, 2 of altitudes, 1 of side
Construct a parallelogram
A
B
C
D
ABCD
A
BC
D
, if three points are marked on the plane: the midpoints of its altitudes
B
H
BH
B
H
and
B
P
BP
BP
and the midpoint of the side
A
D
AD
A
D
.
2
2
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max triangle area, starting with intersecting circles
Two circles intersect at points
P
P
P
and
Q
Q
Q
. Point
A
A
A
lies on the first circle, but outside the second. Lines
A
P
AP
A
P
and
A
Q
AQ
A
Q
intersect the second circle at points
B
B
B
and
C
C
C
, respectively. Indicate the position of point
A
A
A
at which triangle
A
B
C
ABC
A
BC
has the largest area.(D. Prokopenko)
EF = FL. wanted, starting with isosceles right triangle, BK=CL
An isosceles right-angled triangle
A
B
C
ABC
A
BC
is given. On the extensions of sides
A
B
AB
A
B
and
A
C
AC
A
C
, behind vertices
B
B
B
and
C
C
C
equal segments
B
K
BK
B
K
and
C
L
CL
C
L
were laid.
E
E
E
and F are the points of intersection of the segment
K
L
KL
K
L
and the lines perpendicular to the
K
C
KC
K
C
, passing through the points
B
B
B
and
A
A
A
, respectively. Prove that
E
F
=
F
L
EF = FL
EF
=
F
L
.
1
2
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covering a cube os edge 1 with a strip 12x1
Given a rectangular strip of measure
12
×
1
12 \times 1
12
×
1
. Paste this strip in two layers over the cube with edge
1
1
1
(the strip can be bent, but cannot be cut).(V. Shevyakov)
triangle cut in 5 triangles similar to the big, is it right one?
The triangle was divided into five triangles similar to it. Is it true that the original triangle is right-angled?(S. Markelov)