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Problems
Contests
National and Regional Contests
Russia Contests
Oral Moscow Geometry Olympiad
2015 Oral Moscow Geometry Olympiad
2015 Oral Moscow Geometry Olympiad
Part of
Oral Moscow Geometry Olympiad
Subcontests
(6)
6
2
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prove that 2 points are the feet of the altitudes
In the acute-angled non-isosceles triangle
A
B
C
ABC
A
BC
, the height
A
H
AH
A
H
is drawn. Points
B
1
B_1
B
1
and
C
1
C_1
C
1
are marked on the sides
A
C
AC
A
C
and
A
B
AB
A
B
, respectively, so that
H
A
HA
H
A
is the angle bisector of
B
1
H
C
1
B_1HC_1
B
1
H
C
1
and quadrangle
B
C
1
B
1
C
BC_1B_1C
B
C
1
B
1
C
is cyclic. Prove that
B
1
B_1
B
1
and
C
1
C_1
C
1
are feet of the altitudes of triangle
A
B
C
ABC
A
BC
.
3 circumcircles and a circle of given diameter concurrent, altitudes related
In an acute-angled isosceles triangle
A
B
C
ABC
A
BC
, altitudes
C
C
1
CC_1
C
C
1
and
B
B
1
BB_1
B
B
1
intersect the line passing through the vertex
A
A
A
and parallel to the line
B
C
BC
BC
, at points
P
P
P
and
Q
Q
Q
. Let
A
0
A_0
A
0
be the midpoint of side
B
C
BC
BC
, and
A
A
1
AA_1
A
A
1
the altitude. Lines
A
0
C
1
A_0C_1
A
0
C
1
and
A
0
B
1
A_0B_1
A
0
B
1
intersect line
P
Q
PQ
PQ
at points
K
K
K
and
L
L
L
. Prove that the circles circumscribed around triangles
P
Q
A
1
,
K
L
A
0
,
A
1
B
1
C
1
PQA_1, KLA_0, A_1B_1C_1
PQ
A
1
,
K
L
A
0
,
A
1
B
1
C
1
and a circle with a diameter
A
A
1
AA_1
A
A
1
intersect at one point.
5
2
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rhombus BCDE ouside equilateral ABE, AF <BD wanted
On the
B
E
BE
BE
side of a regular
A
B
E
ABE
A
BE
triangle, a
B
C
D
E
BCDE
BC
D
E
rhombus is built outside it. The segments
A
C
AC
A
C
and
B
D
BD
B
D
intersect at point
F
F
F
. Prove that
A
F
<
B
D
AF <BD
A
F
<
B
D
.
prove that planes are tangent to a fixed sphere or pass through a fixed point
A triangle
A
B
C
ABC
A
BC
and spheres are given in space
S
1
S_1
S
1
and
S
2
S_2
S
2
, each of which passes through points
A
,
B
A, B
A
,
B
and
C
C
C
. For points
M
M
M
spheres
S
1
S_1
S
1
not lying in the plane of triangle
A
B
C
ABC
A
BC
are drawn lines
M
A
,
M
B
MA, MB
M
A
,
MB
and
M
C
MC
MC
, intersecting the sphere
S
2
S_2
S
2
for the second time at points
A
1
,
B
1
A_1,B_1
A
1
,
B
1
and
C
1
C_1
C
1
, respectively. Prove that the planes passing through points
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
, touch a fixed sphere or pass through a fixed point.
4
2
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3 incircles inside a trapezoid, angle bisectors of A,D intersec on side BC
In trapezoid
A
B
C
D
ABCD
A
BC
D
, the bisectors of angles
A
A
A
and
D
D
D
intersect at point
E
E
E
lying on the side of
B
C
BC
BC
. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base
A
B
AB
A
B
at the point
K
K
K
, and two others touch the bisector
D
E
DE
D
E
at points
M
M
M
and
N
N
N
. Prove that
B
K
=
M
N
BK = MN
B
K
=
MN
.
equal angles wanted, circumcircle, tangents, midpoints given
In triangle
A
B
C
ABC
A
BC
, point
M
M
M
is the midpoint of
B
C
,
P
BC, P
BC
,
P
is the intersection point of the tangents at points
B
B
B
and
C
C
C
of the circumscribed circle,
N
N
N
is the midpoint of the segment
M
P
MP
MP
. The segment
A
N
AN
A
N
intersects the circumscribed circle at point
Q
Q
Q
. Prove that
∠
P
M
Q
=
∠
M
A
Q
\angle PMQ = \angle MAQ
∠
PMQ
=
∠
M
A
Q
.
3
2
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triangle reconstruction, 3 points given, 3 equal segments related
In triangle
A
B
C
ABC
A
BC
, points
D
,
E
D, E
D
,
E
, and
F
F
F
are marked on sides
A
C
,
B
C
AC, BC
A
C
,
BC
, and
A
B
AB
A
B
respectively, so that
A
D
=
A
B
AD = AB
A
D
=
A
B
,
E
C
=
D
C
EC = DC
EC
=
D
C
,
B
F
=
B
E
BF = BE
BF
=
BE
. After that, they erased everything except points
E
,
F
E, F
E
,
F
and
D
D
D
. Reconstruct the triangle
A
B
C
ABC
A
BC
(no study required).
S_{AOK} = S_{AOB} + S_{DOK}, areas, starting with a trapezoid
O
O
O
is the intersection point of the diagonals of the trapezoid
A
B
C
D
ABCD
A
BC
D
. A line passing through
C
C
C
and a point symmetric to
B
B
B
with respect to
O
O
O
, intersects the base
A
D
AD
A
D
at the point
K
K
K
. Prove that
S
A
O
K
=
S
A
O
B
+
S
D
O
K
S_{AOK} = S_{AOB} + S_{DOK}
S
A
O
K
=
S
A
OB
+
S
D
O
K
.
2
2
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angle wanted, square and equilateral related
The square
A
B
C
D
ABCD
A
BC
D
and the equilateral triangle
M
K
L
MKL
M
K
L
are located as shown in the figure. Find the angle
∠
P
Q
D
\angle PQD
∠
PQ
D
. https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQKgjvzy1WhwkMJbcV_C0iveelYmm75FpaGlWgZ-Ap_uQUiegaKYafelo-J_3rMgKMgpMp5soYc1LVYLI8H4riC6R-f8eq2DiWTGGII08xQkwu7t2KVD4pKX4_IN-gC7DVRhdVZSjbaj2S/s1600/oral+moscow+geometry+2015+8.9+p2.png
midpoint wanted, 3 perpendicular bisectors related
Line
ℓ
\ell
ℓ
is perpendicular to one of the medians of the triangle. The perpendicular bisectors of the sides of this triangle intersect line
ℓ
\ell
ℓ
at three points. Prove that one of them is the midpoint of the segment formed by the remaining two.
1
2
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criterion for equality of trapezoids or not? 2 equal angles and diagonals
Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?
1 altitude bisects 1 median => another altitude bisects another median
In triangle
A
B
C
ABC
A
BC
, the altitude
A
H
AH
A
H
passes through midpoint of the median
B
M
BM
BM
. Prove that in the triangle
B
M
C
BMC
BMC
also one of the altitudes passes through the midpoint of one of the medians.