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Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2009 Sharygin Geometry Olympiad
2009 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(24)
24
1
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A sphere is inscribed into a quadrangular pyramid
A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected to the edges of the base. Prove that these projections are concyclic.
23
1
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regular 2n-gon
Is it true that for each
n
n
n
, the regular
2
n
2n
2
n
-gon is a projection of some polyhedron having not greater than n \plus{} 2 faces?
22
1
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Construct a bicentric quadrilateral
Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.
21
1
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center of parallelogram
The opposite sidelines of quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at points
P
P
P
and
Q
Q
Q
. Two lines passing through these points meet the side of
A
B
C
D
ABCD
A
BC
D
in four points which are the vertices of a parallelogram. Prove that the center of this parallelogram lies on the line passing through the midpoints of diagonals of
A
B
C
D
ABCD
A
BC
D
.
20
1
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nice concyclic
Suppose
H
H
H
and
O
O
O
are the orthocenter and the circumcenter of acute triangle
A
B
C
ABC
A
BC
;
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
and
C
C
1
CC_1
C
C
1
are the altitudes of the triangle. Point
C
2
C_2
C
2
is the reflection of
C
C
C
in
A
1
B
1
A_1B_1
A
1
B
1
. Prove that
H
H
H
,
O
O
O
,
C
1
C_1
C
1
and
C
2
C_2
C
2
are concyclic.
19
1
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Find the maximal and the minimal value of k
Given convex
n
n
n
-gon
A
1
…
A
n
A_1\ldots A_n
A
1
…
A
n
. Let
P
i
P_i
P
i
( i \equal{} 1,\ldots , n) be such points on its boundary that
A
i
P
i
A_iP_i
A
i
P
i
bisects the area of polygon. All points
P
i
P_i
P
i
don't coincide with any vertex and lie on
k
k
k
sides of
n
n
n
-gon. What is the maximal and the minimal value of
k
k
k
for each given
n
n
n
?
18
1
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Find the locus of incenters
Given three parallel lines on the plane. Find the locus of incenters of triangles with vertices lying on these lines (a single vertex on each line).
17
1
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a concurrency problem
Given triangle
A
B
C
ABC
A
BC
and two points
X
X
X
,
Y
Y
Y
not lying on its circumcircle. Let
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
be the projections of
X
X
X
to
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
, and
A
2
A_2
A
2
,
B
2
B_2
B
2
,
C
2
C_2
C
2
be the projections of
Y
Y
Y
. Prove that the perpendiculars from
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
to
B
2
C
2
B_2C_2
B
2
C
2
,
C
2
A
2
C_2A_2
C
2
A
2
,
A
2
B
2
A_2B_2
A
2
B
2
, respectively, concur if and only if line
X
Y
XY
X
Y
passes through the circumcenter of
A
B
C
ABC
A
BC
.
16
1
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prove an angle is right
Three lines passing through point
O
O
O
form equal angles by pairs. Points
A
1
A_1
A
1
,
A
2
A_2
A
2
on the first line and
B
1
B_1
B
1
,
B
2
B_2
B
2
on the second line are such that the common point
C
1
C_1
C
1
of
A
1
B
1
A_1B_1
A
1
B
1
and
A
2
B
2
A_2B_2
A
2
B
2
lies on the third line. Let
C
2
C_2
C
2
be the common point of
A
1
B
2
A_1B_2
A
1
B
2
and
A
2
B
1
A_2B_1
A
2
B
1
. Prove that angle
C
1
O
C
2
C_1OC_2
C
1
O
C
2
is right.
15
1
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prove triangle of maximal area is isosceles
Given a circle and a point
C
C
C
not lying on this circle. Consider all triangles
A
B
C
ABC
A
BC
such that points
A
A
A
and
B
B
B
lie on the given circle. Prove that the triangle of maximal area is isosceles.
14
1
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Determine the area of triangle
Given triangle
A
B
C
ABC
A
BC
of area 1. Let
B
M
BM
BM
be the perpendicular from
B
B
B
to the bisector of angle
C
C
C
. Determine the area of triangle
A
M
C
AMC
A
MC
.
13
1
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A construction problem
In triangle
A
B
C
ABC
A
BC
, one has marked the incenter, the foot of altitude from vertex
C
C
C
and the center of the excircle tangent to side
A
B
AB
A
B
. After this, the triangle was erased. Restore it.
12
1
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Equal angles again
Let
C
L
CL
C
L
be a bisector of triangle
A
B
C
ABC
A
BC
. Points
A
1
A_1
A
1
and
B
1
B_1
B
1
are the reflections of
A
A
A
and
B
B
B
in
C
L
CL
C
L
, points
A
2
A_2
A
2
and
B
2
B_2
B
2
are the reflections of
A
A
A
and
B
B
B
in
L
L
L
. Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the circumcenters of triangles
A
B
1
B
2
AB_1B_2
A
B
1
B
2
and
B
A
1
A
2
BA_1A_2
B
A
1
A
2
respectively. Prove that angles
O
1
C
A
O_1CA
O
1
C
A
and
O
2
C
B
O_2CB
O
2
CB
are equal.
11
1
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Given a quadrilateral
Given quadrilateral
A
B
C
D
ABCD
A
BC
D
. The circumcircle of
A
B
C
ABC
A
BC
is tangent to side
C
D
CD
C
D
, and the circumcircle of
A
C
D
ACD
A
C
D
is tangent to side
A
B
AB
A
B
. Prove that the length of diagonal
A
C
AC
A
C
is less than the distance between the midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
.
10
1
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Determine angle C
Let
A
B
C
ABC
A
BC
be an acute triangle,
C
C
1
CC_1
C
C
1
its bisector,
O
O
O
its circumcenter. The perpendicular from
C
C
C
to
A
B
AB
A
B
meets line
O
C
1
OC_1
O
C
1
in a point lying on the circumcircle of
A
O
B
AOB
A
OB
. Determine angle
C
C
C
.
9
1
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n points on the plane
Given
n
n
n
points on the plane, which are the vertices of a convex polygon,
n
>
3
n > 3
n
>
3
. There exists
k
k
k
regular triangles with the side equal to
1
1
1
and the vertices at the given points. [*] Prove that
k
<
2
3
n
k < \frac {2}{3}n
k
<
3
2
n
. [*] Construct the configuration with
k
>
0.666
n
k > 0.666n
k
>
0.666
n
.
8
4
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7
4
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6
4
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5
4
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4
4
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3
4
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2
4
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1
4
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