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Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2007 Sharygin Geometry Olympiad
2007 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(21)
1
4
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2
4
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18
1
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locus of vertices of triangles with given orthocenter and circumcenter
Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.
21
1
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area of surface of a chamber by 3 pipes
There are two pipes on the plane (the pipes are circular cylinders of equal size,
4
4
4
m around). Two of them are parallel and, being tangent one to another in the common generatrix, form a tunnel over the plane. The third pipe is perpendicular to two others and cuts out a chamber in the tunnel. Determine the area of the surface of this chamber.
20
1
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max volume of a pyramid with base an equilateral triangle and 2 right angles
The base of a pyramid is a regular triangle having side of size
1
1
1
. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.
19
1
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min angle so that S_{PAQ}<S_{BMC}, related to cyclic quadrilateral
Into an angle
A
A
A
of size
a
a
a
, a circle is inscribed tangent to its sides at points
B
B
B
and
C
C
C
. A line tangent to this circle at a point M meets the segments
A
B
AB
A
B
and
A
C
AC
A
C
at points
P
P
P
and
Q
Q
Q
respectively. What is the minimum
a
a
a
such that the inequality
S
P
A
Q
<
S
B
M
C
S_{PAQ}<S_{BMC}
S
P
A
Q
<
S
BMC
is possible?
17
1
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triangles can be cut into 3 triangles having equal radii of circumcircles
What triangles can be cut into three triangles having equal radii of circumcircles?
16
1
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starting with 2 points on sides of an angle, midpoints given and wanted
On two sides of an angle, points
A
,
B
A, B
A
,
B
are chosen. The midpoint
M
M
M
of the segment
A
B
AB
A
B
belongs to two lines such that one of them meets the sides of the angle at points
A
1
,
B
1
A_1, B_1
A
1
,
B
1
, and the other at points
A
2
,
B
2
A_2, B_2
A
2
,
B
2
. The lines
A
1
B
2
A_1B_2
A
1
B
2
and
A
2
B
1
A_2B_1
A
2
B
1
meet
A
B
AB
A
B
at points
P
P
P
and
Q
Q
Q
. Prove that
M
M
M
is the midpoint of
P
Q
PQ
PQ
.
15
1
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equal angles starting with angles bisectors
In a triangle
A
B
C
ABC
A
BC
, let
A
A
′
,
B
B
′
AA', BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
be the bisectors. Suppose
A
′
B
′
∩
C
C
′
=
P
A'B' \cap CC' =P
A
′
B
′
∩
C
C
′
=
P
and
A
′
C
′
∩
B
B
′
=
Q
A'C' \cap BB'= Q
A
′
C
′
∩
B
B
′
=
Q
. Prove that
∠
P
A
C
=
∠
Q
A
B
\angle PAC = \angle QAB
∠
P
A
C
=
∠
Q
A
B
.
14
1
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equal angles lead to equal ones, related to midpoints of diagonals of trapezoid
In a trapezium with bases
A
D
AD
A
D
and
B
C
BC
BC
, let
P
P
P
and
Q
Q
Q
be the middles of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
respectively. Prove that if
∠
D
A
Q
=
∠
C
A
B
\angle DAQ = \angle CAB
∠
D
A
Q
=
∠
C
A
B
then
∠
P
B
A
=
∠
D
B
C
\angle PBA = \angle DBC
∠
PB
A
=
∠
D
BC
.
13
1
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fixed point starting with equal segments on a side of triangle
On the side
A
B
AB
A
B
of a triangle
A
B
C
ABC
A
BC
, two points
X
,
Y
X, Y
X
,
Y
are chosen so that
A
X
=
B
Y
AX = BY
A
X
=
B
Y
. Lines
C
X
CX
CX
and
C
Y
CY
C
Y
meet the circumcircle of the triangle, for the second time, at points
U
U
U
and
V
V
V
. Prove that all lines
U
V
UV
U
V
(for all
X
,
Y
X, Y
X
,
Y
, given
A
,
B
,
C
A, B, C
A
,
B
,
C
) have a common point.
12
1
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perpendicularity related to a rectangle and a random point
A rectangle
A
B
C
D
ABCD
A
BC
D
and a point
P
P
P
are given. Lines passing through
A
A
A
and
B
B
B
and perpendicular to
P
C
PC
PC
and
P
D
PD
P
D
respectively, meet at a point
Q
Q
Q
. Prove that
P
Q
⊥
A
B
PQ \perp AB
PQ
⊥
A
B
.
11
1
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ratio of visible distances of a boy on ground and on his father’s shoulders,=
A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of
1
1
1
m above sea-level, and if he seats on father’s shoulders, they are at a height of
2
2
2
m. What is the ratio of distances visible for him in two eases? (Find the answer to
0
,
1
0,1
0
,
1
, assuming that the radius of Earth equals
6000
6000
6000
km.)
10
1
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locus of centers of equilateral triangles, 3 points lie on 3 lines of triangle
Find the locus of centers of regular triangles such that three given points
A
,
B
,
C
A, B, C
A
,
B
,
C
lie respectively on three lines containing sides of the triangle.
9
1
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angle chasing in 2 convex quadrangles
Suppose two convex quadrangles are such that the sides of each of them lie on the perpendicular bisectors of the sides of the other one. Determine their angles,
8
1
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given 3 concurrent circles and 3 collinear intersections, equal triangles wanted
Three circles pass through a point
P
P
P
, and the second points of their intersection
A
,
B
,
C
A, B, C
A
,
B
,
C
lie on a straight line. Let
A
1
B
1
,
C
1
A_1 B_1, C_1
A
1
B
1
,
C
1
be the second meets of lines
A
P
,
B
P
,
C
P
AP, BP, CP
A
P
,
BP
,
CP
with the corresponding circles. Let
C
2
C_2
C
2
be the intersections of lines
A
B
1
AB_1
A
B
1
and
B
A
1
BA_1
B
A
1
. Let
A
2
,
B
2
A_2, B_2
A
2
,
B
2
be defined similarly. 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 equal,
7
1
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tangential convex polygon, touchpoints create polygon with same angles, regular?
A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?
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4
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4
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4
4
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3
4
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