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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2006 Oral Moscow Geometry Olympiad
2006 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
6
2
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equilateral from line of circumcenter and centroid, <A=60^o
In an acute-angled triangle, one of the angles is
6
0
o
60^o
6
0
o
. Prove that the line passing through the center of the circumcircle and the intersection point of the medians of the triangle cuts off an equilateral triangle from it.(A. Zaslavsky)
similar triangles wanted, cevians, circumcircle, symmetric points wrt sides
Given triangle
A
B
C
ABC
A
BC
and points
P
P
P
. Let
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
be the second points of intersection of straight lines
A
P
,
B
P
,
C
P
AP, BP, CP
A
P
,
BP
,
CP
with the circumscribed circle of
A
B
C
ABC
A
BC
. Let points
A
2
,
B
2
,
C
2
A_2, B_2, C_2
A
2
,
B
2
,
C
2
be symmetric to
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
wrt
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively. Prove that the triangles
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
and
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
are similar.(A. Zaslavsky)
5
2
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midpoint wanted, equilateral triangles on sides of a triangle related
Equilateral triangles
A
B
C
1
,
B
C
A
1
,
C
A
B
1
ABC_1, BCA_1, CAB_1
A
B
C
1
,
BC
A
1
,
C
A
B
1
are built on the sides of the triangle
A
B
C
ABC
A
BC
to the outside. On the segment
A
1
B
1
A_1B_1
A
1
B
1
to the outer side of the triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
, an equilateral triangle
A
1
B
1
C
2
A_1B_1C_2
A
1
B
1
C
2
is constructed. Prove that
C
C
C
is the midpoint of the segment
C
1
C
2
C_1C_2
C
1
C
2
.(A. Zaslavsky)
exists cyclic quadrilateral, section of a pyramid with convex quadr. base?
The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle?(M. Volchkevich)
4
2
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construct line passing through B so that inradii of 2 small triangles are equal
An arbitrary triangle
A
B
C
ABC
A
BC
is given. Construct a straight line passing through vertex
B
B
B
and dividing it into two triangles, the radii of the inscribed circles of which are equal.(M. Volchkevich)
equal products of distances,cyclic ABCD, tangents concurrent with symmetric line
The quadrangle
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle, the center
O
O
O
of which lies inside it. The tangents to the circle at points
A
A
A
and
C
C
C
and a straight line, symmetric to
B
D
BD
B
D
wrt point
O
O
O
, intersect at one point. Prove that the products of the distances from
O
O
O
to opposite sides of the quadrilateral are equal.(A. Zaslavsky)
3
2
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A'B' > diameter of incircle, <A'C'B'=90^o
On the sides
A
B
,
B
C
AB, BC
A
B
,
BC
and
A
C
AC
A
C
of the triangle
A
B
C
ABC
A
BC
, points
C
′
,
A
′
C', A'
C
′
,
A
′
and
B
′
B'
B
′
are selected, respectively, so that the angle
A
′
C
′
B
′
A'C'B'
A
′
C
′
B
′
is right. Prove that the segment
A
′
B
′
A'B'
A
′
B
′
is longer than the diameter of the inscribed circle of the triangle
A
B
C
ABC
A
BC
.(M. Volchkevich)
locus of centroids of triangles, 2 vertices on one circle, 3rd on another
Two non-rolling circles
C
1
C_1
C
1
and
C
2
C_2
C
2
with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
and radii
2
R
2R
2
R
and
R
R
R
, respectively, are given on the plane. Find the locus of the centers of gravity of triangles in which one vertex lies on
C
1
C_1
C
1
and the other two lie on
C
2
C_2
C
2
. (B. Frenkin)
2
2
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ratio of sidelengths of rectangle circumscribed in L shape
Determine the ratio of the sides of the rectangle circumscribed around a corner of five cells (see figure).(M. Evdokimov) https://cdn.artofproblemsolving.com/attachments/f/f/9c3e345f33cabbbd83f65d7240aac29a163b19.png
any 3 out of 6 segments form a triangle, what about a tetrahedron?
Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron?(S. Markelov)
1
2
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tangent at intersection of diagonals of a cyclic ABD parallel to a side
The diagonals of the inscribed quadrangle
A
B
C
D
ABCD
A
BC
D
intersect at point
K
K
K
. Prove that the tangent at point
K
K
K
to the circle circumscribed around the triangle
A
B
K
ABK
A
B
K
is parallel to
C
D
CD
C
D
.(A Zaslavsky)
construct line that divides a triangle into 2 polygons of equal circumradii
An arbitrary triangle
A
B
C
ABC
A
BC
is given. Construct a line that divides it into two polygons, which have equal radii of the circumscribed circles.(L. Blinkov)