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Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2010 Sharygin Geometry Olympiad
2010 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(25)
25
1
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Ratio of the edges of the icosahedrons (25)
For two different regular icosahedrons it is known that some six of their vertices are vertices of a regular octahedron. Find the ratio of the edges of these icosahedrons.
24
1
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Find the locus of points Y (24)
Let us have a line
ℓ
\ell
ℓ
in the space and a point
A
A
A
not lying on
ℓ
.
\ell.
ℓ
.
For an arbitrary line
ℓ
′
\ell'
ℓ
′
passing through
A
A
A
,
X
Y
XY
X
Y
(
Y
Y
Y
is on
ℓ
′
\ell'
ℓ
′
) is a common perpendicular to the lines
ℓ
\ell
ℓ
and
ℓ
′
.
\ell'.
ℓ
′
.
Find the locus of points
Y
.
Y.
Y
.
23
1
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Prove that the lines AD, BE and CF are concurrent (23)
A cyclic hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is such that
A
B
⋅
C
F
=
2
B
C
⋅
F
A
,
C
D
⋅
E
B
=
2
D
E
⋅
B
C
AB \cdot CF= 2BC \cdot FA, CD \cdot EB = 2 DE \cdot BC
A
B
⋅
CF
=
2
BC
⋅
F
A
,
C
D
⋅
EB
=
2
D
E
⋅
BC
and
E
F
⋅
A
D
=
2
F
A
⋅
D
E
.
EF \cdot AD = 2FA \cdot DE.
EF
⋅
A
D
=
2
F
A
⋅
D
E
.
Prove that the lines
A
D
,
B
E
AD, BE
A
D
,
BE
and
C
F
CF
CF
are concurrent.
22
1
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Prove that there exist four points A, B, C, D (22)
A circle centered at a point
F
F
F
and a parabola with focus
F
F
F
have two common points. Prove that there exist four points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
on the circle such that the lines
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
touch the parabola.
21
1
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Prove that S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC} (21)
A given convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is such that
∠
A
B
D
+
∠
A
C
D
>
∠
B
A
C
+
∠
B
D
C
.
\angle ABD + \angle ACD > \angle BAC + \angle BDC.
∠
A
B
D
+
∠
A
C
D
>
∠
B
A
C
+
∠
B
D
C
.
Prove that
S
A
B
D
+
S
A
C
D
>
S
B
A
C
+
S
B
D
C
.
S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.
S
A
B
D
+
S
A
C
D
>
S
B
A
C
+
S
B
D
C
.
20
1
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Common point of AA' and BB' (20)
The incircle of an acute-angled triangle
A
B
C
ABC
A
BC
touches
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
at points
C
1
,
A
1
,
B
1
C_1, A_1, B_1
C
1
,
A
1
,
B
1
respectively. Points
A
2
,
B
2
A_2, B_2
A
2
,
B
2
are the midpoints of the segments
B
1
C
1
,
A
1
C
1
B_1C_1, A_1C_1
B
1
C
1
,
A
1
C
1
respectively. Let
P
P
P
be a common point of the incircle and the line
C
O
CO
CO
, where
O
O
O
is the circumcenter of triangle
A
B
C
.
ABC.
A
BC
.
Let also
A
′
A'
A
′
and
B
′
B'
B
′
be the second common points of
P
A
2
PA_2
P
A
2
and
P
B
2
PB_2
P
B
2
with the incircle. Prove that a common point of
A
A
′
AA'
A
A
′
and
B
B
′
BB'
B
B
′
lies on the altitude of the triangle dropped from the vertex
C
.
C.
C
.
19
1
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Prove that XV=YU (19)
A quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed into a circle with center
O
.
O.
O
.
Points
P
P
P
and
Q
Q
Q
are opposite to
C
C
C
and
D
D
D
respectively. Two tangents drawn to that circle at these points meet the line
A
B
AB
A
B
in points
E
E
E
and
F
.
F.
F
.
(
A
A
A
is between
E
E
E
and
B
B
B
,
B
B
B
is between
A
A
A
and
F
F
F
). The line
E
O
EO
EO
meets
A
C
AC
A
C
and
B
C
BC
BC
in points
X
X
X
and
Y
Y
Y
respectively, and the line
F
O
FO
FO
meets
A
D
AD
A
D
and
B
D
BD
B
D
in points
U
U
U
and
V
V
V
respectively. Prove that
X
V
=
Y
U
.
XV=YU.
X
V
=
Y
U
.
18
1
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Prove that FG passes through the midpoint of AC (18)
A point
B
B
B
lies on a chord
A
C
AC
A
C
of circle
ω
.
\omega.
ω
.
Segments
A
B
AB
A
B
and
B
C
BC
BC
are diameters of circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
centered at
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively. These circles intersect
ω
\omega
ω
for the second time in points
D
D
D
and
E
E
E
respectively. The rays
O
1
D
O_1D
O
1
D
and
O
2
E
O_2E
O
2
E
meet in a point
F
,
F,
F
,
and the rays
A
D
AD
A
D
and
C
E
CE
CE
do in a point
G
.
G.
G
.
Prove that the line
F
G
FG
FG
passes through the midpoint of the segment
A
C
.
AC.
A
C
.
17
1
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Construct a triangle (17)
Construct a triangle, if the lengths of the bisectrix and of the altitude from one vertex, and of the median from another vertex are given.
16
1
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XC passes through the midpoint of the segment (16)
A circle touches the sides of an angle with vertex
A
A
A
at points
B
B
B
and
C
.
C.
C
.
A line passing through
A
A
A
intersects this circle in points
D
D
D
and
E
.
E.
E
.
A chord
B
X
BX
BX
is parallel to
D
E
.
DE.
D
E
.
Prove that
X
C
XC
XC
passes through the midpoint of the segment
D
E
.
DE.
D
E
.
15
1
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The sum of diameters is equal to BC (15)
Let
A
A
1
,
B
B
1
AA_1, BB_1
A
A
1
,
B
B
1
and
C
C
1
CC_1
C
C
1
be the altitudes of an acute-angled triangle
A
B
C
.
ABC.
A
BC
.
A
A
1
AA_1
A
A
1
meets
B
1
C
1
B_1C_1
B
1
C
1
in a point
K
.
K.
K
.
The circumcircles of triangles
A
1
K
C
1
A_1KC_1
A
1
K
C
1
and
A
1
K
B
1
A_1KB_1
A
1
K
B
1
intersect the lines
A
B
AB
A
B
and
A
C
AC
A
C
for the second time at points
N
N
N
and
L
L
L
respectively. Prove thata) The sum of diameters of these two circles is equal to
B
C
,
BC,
BC
,
b)
A
1
N
B
B
1
+
A
1
L
C
C
1
=
1.
\frac{A_1N}{BB_1} + \frac{A_1L}{CC_1}=1.
B
B
1
A
1
N
+
C
C
1
A
1
L
=
1.
14
1
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Inequality on area of a triangnle and a quadrilateral (14)
We have a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
and a point
M
M
M
on its side
A
D
AD
A
D
such that
C
M
CM
CM
and
B
M
BM
BM
are parallel to
A
B
AB
A
B
and
C
D
CD
C
D
respectively. Prove that
S
A
B
C
D
≥
3
S
B
C
M
.
S_{ABCD} \geq 3 S_{BCM}.
S
A
BC
D
≥
3
S
BCM
.
Remark.
S
S
S
denotes the area function.
13
1
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Prove that AK/KC = AD/CD (13)
Let us have a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
such that
A
B
=
B
C
.
AB=BC.
A
B
=
BC
.
A point
K
K
K
lies on the diagonal
B
D
,
BD,
B
D
,
and
∠
A
K
B
+
∠
B
K
C
=
∠
A
+
∠
C
.
\angle AKB+\angle BKC=\angle A + \angle C.
∠
A
K
B
+
∠
B
K
C
=
∠
A
+
∠
C
.
Prove that
A
K
⋅
C
D
=
K
C
⋅
A
D
.
AK \cdot CD = KC \cdot AD.
A
K
⋅
C
D
=
K
C
⋅
A
D
.
12
1
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Prove that B'M is parallel to BC and AK is tangent (12)
Let
A
C
AC
A
C
be the greatest leg of a right triangle
A
B
C
,
ABC,
A
BC
,
and
C
H
CH
C
H
be the altitude to its hypotenuse. The circle of radius
C
H
CH
C
H
centered at
H
H
H
intersects
A
C
AC
A
C
in point
M
.
M.
M
.
Let a point
B
′
B'
B
′
be the reflection of
B
B
B
with respect to the point
H
.
H.
H
.
The perpendicular to
A
B
AB
A
B
erected at
B
′
B'
B
′
meets the circle in a point
K
K
K
. Prove thata)
B
′
M
∥
B
C
B'M \parallel BC
B
′
M
∥
BC
b)
A
K
AK
A
K
is tangent to the circle.
11
1
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Find all possible values of n - convex n-gon (11)
A convex
n
−
n-
n
−
gon is split into three convex polygons. One of them has
n
n
n
sides, the second one has more than
n
n
n
sides, the third one has less than
n
n
n
sides. Find all possible values of
n
.
n.
n
.
10
1
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Find a point using a ruler and drawing at most 8 lines (10)
Let three lines forming a triangle
A
B
C
ABC
A
BC
be given. Using a two-sided ruler and drawing at most eight lines construct a point
D
D
D
on the side
A
B
AB
A
B
such that
A
D
B
D
=
B
C
A
C
.
\frac{AD}{BD}=\frac{BC}{AC}.
B
D
A
D
=
A
C
BC
.
9
1
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Find all possible values of number of good points (9)
A point inside a triangle is called "good" if three cevians passing through it are equal. Assume for an isosceles triangle
A
B
C
(
A
B
=
B
C
)
ABC \ (AB=BC)
A
BC
(
A
B
=
BC
)
the total number of "good" points is odd. Find all possible values of this number.
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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