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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2005 Oral Moscow Geometry Olympiad
2005 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
6
2
Hide problems
perpendicular wanted, incenter, midpoints related
Let
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
are the midpoints of the sides of the triangle
A
B
C
,
I
ABC, I
A
BC
,
I
is the center of the circle inscribed in it. Let
C
2
C_2
C
2
be the intersection point of lines
C
1
I
C_1 I
C
1
I
and
A
1
B
1
A_1B_1
A
1
B
1
. Let
C
3
C_3
C
3
be the intersection point of lines
C
C
2
CC_2
C
C
2
and
A
B
AB
A
B
. Prove that line
I
C
3
IC_3
I
C
3
is perpendicular to line
A
B
AB
A
B
.(A. Zaslavsky)
for any 3 of 6 lines, exists a 4th such that they are tangent to a circle
Six straight lines are drawn on the plane. It is known that for any three of them there is a fourth of the same set of lines, such that all four will touch some circle. Do all six lines necessarily touch the same circle? (I. Bogdanov)
5
2
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point construction, intersections of cevians with circumcircle, equilateral
The triangle
A
B
C
ABC
A
BC
is inscribed in the circle. Construct a point
P
P
P
such that the points of intersection of lines
A
P
,
B
P
AP, BP
A
P
,
BP
and
C
P
CP
CP
with this circle are the vertices of an equilateral triangle.(A. Zaslavsky)
MA + MB + MC <=max (AB + BC, BC + AC, AC + AB) for interior M in ABC
An arbitrary point
M
M
M
is chosen inside the triangle
A
B
C
ABC
A
BC
. Prove that
M
A
+
M
B
+
M
C
≤
m
a
x
(
A
B
+
B
C
,
B
C
+
A
C
,
A
C
+
A
B
)
MA + MB + MC \le max (AB + BC, BC + AC, AC + AB)
M
A
+
MB
+
MC
≤
ma
x
(
A
B
+
BC
,
BC
+
A
C
,
A
C
+
A
B
)
.(N. Sedrakyan)
4
2
Hide problems
FD _|_ BE wanted, hexagon ABCDE, AB = BC, CD = DE, EF = FA, <A=<C=90^o
Given a hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
, in which
A
B
=
B
C
,
C
D
=
D
E
,
E
F
=
F
A
AB = BC, CD = DE, EF = FA
A
B
=
BC
,
C
D
=
D
E
,
EF
=
F
A
, and angles
A
A
A
and
C
C
C
are right. Prove that lines
F
D
FD
F
D
and
B
E
BE
BE
are perpendicular.(B. Kukushkin)
aphere inscribed into pyramid, with parallelogram base
A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal.(M. Volchkevich)
3
2
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KN + LM >= AC wanted, AK = BL, CN = BM
In triangle
A
B
C
ABC
A
BC
, points
K
,
P
K ,P
K
,
P
are chosen on the side
A
B
AB
A
B
so that
A
K
=
B
L
AK = BL
A
K
=
B
L
, and points
M
,
N
M,N
M
,
N
are chosen on the side
B
C
BC
BC
so that
C
N
=
B
M
CN = BM
CN
=
BM
. Prove that
K
N
+
L
M
≥
A
C
KN + LM \ge AC
K
N
+
L
M
≥
A
C
.(I. Bogdanov)
cover plane with specific non convex pentagons
A
B
C
B
E
ABCBE
A
BCBE
is a regular pentagon. Point
B
′
B'
B
′
is symmetric to point
B
B
B
wrt line
A
C
AC
A
C
(see figure). Is it possible to pave the plane with pentagons equal to
A
B
′
C
B
E
AB'CBE
A
B
′
CBE
?(S. Markelov) https://cdn.artofproblemsolving.com/attachments/9/2/cbb5756517e85e56c4a931e761a6b4da8fe547.png
2
2
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equal angles wanted inside a parallelogram, parallel, angle bisectors
A parallelogram of
A
B
C
D
ABCD
A
BC
D
is given. Line parallel to
A
B
AB
A
B
intersects the bisectors of angles
A
A
A
and
C
C
C
at points
P
P
P
and
Q
Q
Q
, respectively. Prove that the angles
A
D
P
ADP
A
D
P
and
A
B
Q
ABQ
A
BQ
are equal. (A. Hakobyan)
2 projections and a chord's midpoint collinear wanted
On a circle with diameter
A
B
AB
A
B
, lie points
C
C
C
and
D
D
D
.
X
Y
XY
X
Y
is the diameter passing through the midpoint
K
K
K
of the chord
C
D
CD
C
D
. Point
M
M
M
is the projection of point
X
X
X
onto line
A
C
AC
A
C
, and point
N
N
N
is the projection of point
Y
Y
Y
on line
B
D
BD
B
D
. Prove that points
M
,
N
M, N
M
,
N
and
K
K
K
are collinear.(A. Zaslavsky)
1
2
Hide problems
area bisector in a figure consisted of two joined rectangles
The hexagon has five
9
0
o
90^o
9
0
o
angles and one
27
0
o
270^o
27
0
o
angle (see picture). Use a straight-line ruler to divide it into two equal-sized polygons. https://cdn.artofproblemsolving.com/attachments/d/8/cdd4df68644bb8e04adbe1b265039b82a5382b.png
min circumradii wanted, line // BC in triangle ABC
Given an acute-angled triangle
A
B
C
ABC
A
BC
. A straight line parallel to
B
C
BC
BC
intersects sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
M
M
M
and
P
P
P
, respectively. At what location of the points
M
M
M
and
P
P
P
will the radius of the circle circumscribed about the triangle
B
M
P
BMP
BMP
be the smallest?(I. Sharygin)