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Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2013 Sharygin Geometry Olympiad
2013 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(23)
23
1
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Symmetry planes of convex polytopes
Two convex polytopes
A
A
A
and
B
B
B
do not intersect. The polytope
A
A
A
has exactly
2012
2012
2012
planes of symmetry. What is the maximal number of symmetry planes of the union of
A
A
A
and
B
B
B
, if
B
B
B
has a)
2012
2012
2012
, b)
2013
2013
2013
symmetry planes?c) What is the answer to the question of p.b), if the symmetry planes are replaced by the symmetry axes?
22
1
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Mutually orthogonal perpendiculars
The common perpendiculars to the opposite sidelines of a nonplanar quadrilateral are mutually orthogonal. Prove that they intersect.
20
1
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Common point of all lines C1C2
Let
C
1
C_1
C
1
be an arbitrary point on the side
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
. Points
A
1
A_1
A
1
and
B
1
B_1
B
1
on the rays
B
C
BC
BC
and
A
C
AC
A
C
are such that
∠
A
C
1
B
1
=
∠
B
C
1
A
1
=
∠
A
C
B
\angle AC_1B_1 = \angle BC_1A_1 = \angle ACB
∠
A
C
1
B
1
=
∠
B
C
1
A
1
=
∠
A
CB
. The lines
A
A
1
AA_1
A
A
1
and
B
B
1
BB_1
B
B
1
meet in point
C
2
C_2
C
2
. Prove that all the lines
C
1
C
2
C_1C_2
C
1
C
2
have a common point.
19
1
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Coincidence of concurrency points
a) The incircle of a triangle
A
B
C
ABC
A
BC
touches
A
C
AC
A
C
and
A
B
AB
A
B
at points
B
0
B_0
B
0
and
C
0
C_0
C
0
respectively. The bisectors of angles
B
B
B
and
C
C
C
meet the perpendicular bisector to the bisector
A
L
AL
A
L
in points
Q
Q
Q
and
P
P
P
respectively. Prove that the lines
P
C
0
,
Q
B
0
PC_0, QB_0
P
C
0
,
Q
B
0
and
B
C
BC
BC
concur.b) Let
A
L
AL
A
L
be the bisector of a triangle
A
B
C
ABC
A
BC
. Points
O
1
O_1
O
1
and
O
2
O_2
O
2
are the circumcenters of triangles
A
B
L
ABL
A
B
L
and
A
C
L
ACL
A
C
L
respectively. Points
B
1
B_1
B
1
and
C
1
C_1
C
1
are the projections of
C
C
C
and
B
B
B
to the bisectors of angles
B
B
B
and
C
C
C
respectively. Prove that the lines
O
1
C
1
,
O
2
B
1
,
O_1C_1, O_2B_1,
O
1
C
1
,
O
2
B
1
,
and
B
C
BC
BC
concur.c) Prove that the two points obtained in pp. a) and b) coincide.
18
1
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Prove that angle BAX = angle CAY
Let
A
D
AD
A
D
be a bisector of triangle
A
B
C
ABC
A
BC
. Points
M
M
M
and
N
N
N
are projections of
B
B
B
and
C
C
C
respectively to
A
D
AD
A
D
. The circle with diameter
M
N
MN
MN
intersects
B
C
BC
BC
at points
X
X
X
and
Y
Y
Y
. Prove that
∠
B
A
X
=
∠
C
A
Y
\angle BAX = \angle CAY
∠
B
A
X
=
∠
C
A
Y
.
17
1
Hide problems
Acute angles formed by diagonals of quadrilateral
An acute angle between the diagonals of a cyclic quadrilateral is equal to
ϕ
\phi
ϕ
. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than
ϕ
\phi
ϕ
.
13
1
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Orthocenter collinear iff orthogonal
Let
A
1
A_1
A
1
and
C
1
C_1
C
1
be the tangency points of the incircle of triangle
A
B
C
ABC
A
BC
with
B
C
BC
BC
and
A
B
AB
A
B
respectively,
A
′
A'
A
′
and
C
′
C'
C
′
be the tangency points of the excircle inscribed into the angle
B
B
B
with the extensions of
B
C
BC
BC
and
A
B
AB
A
B
respectively. Prove that the orthocenter
H
H
H
of triangle
A
B
C
ABC
A
BC
lies on
A
1
C
1
A_1C_1
A
1
C
1
if and only if the lines
A
′
C
1
A'C_1
A
′
C
1
and
B
A
BA
B
A
are orthogonal.
12
1
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Distinguishing feet of altitudes and bisectors
On each side of triangle
A
B
C
ABC
A
BC
, two distinct points are marked. It is known that these points are the feet of the altitudes and of the bisectors.a) Using only a ruler determine which points are the feet of the altitudes and which points are the feet of the bisectors.b) Solve p.a) drawing only three lines.
11
1
Hide problems
Inequalities in convex quadrilaterals
a) Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and
r
1
≤
r
2
≤
r
3
≤
r
4
r_1 \le r_2 \le r_3 \le r_4
r
1
≤
r
2
≤
r
3
≤
r
4
be the radii of the incircles of triangles
A
B
C
,
B
C
D
,
C
D
A
,
D
A
B
ABC, BCD, CDA, DAB
A
BC
,
BC
D
,
C
D
A
,
D
A
B
. Can the inequality
r
4
>
2
r
3
r_4 > 2r_3
r
4
>
2
r
3
hold?b) The diagonals of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
meet in point
E
E
E
. Let
r
1
≤
r
2
≤
r
3
≤
r
4
r_1 \le r_2 \le r_3 \le r_4
r
1
≤
r
2
≤
r
3
≤
r
4
be the radii of the incircles of triangles
A
B
E
,
B
C
E
,
C
D
E
,
D
A
E
ABE, BCE, CDE, DAE
A
BE
,
BCE
,
C
D
E
,
D
A
E
. Can the inequality
r
2
>
2
r
1
r_2 > 2r_1
r
2
>
2
r
1
hold?
10
1
Hide problems
Concurrency of lines connecting points of tangencies
The incircle of triangle
A
B
C
ABC
A
BC
touches the side
A
B
AB
A
B
at point
C
′
C'
C
′
; the incircle of triangle
A
C
C
′
ACC'
A
C
C
′
touches the sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
C
1
,
B
1
C_1, B_1
C
1
,
B
1
; the incircle of triangle
B
C
C
′
BCC'
BC
C
′
touches the sides
A
B
AB
A
B
and
B
C
BC
BC
at points
C
2
C_2
C
2
,
A
2
A_2
A
2
. Prove that the lines
B
1
C
1
B_1C_1
B
1
C
1
,
A
2
C
2
A_2C_2
A
2
C
2
, and
C
C
′
CC'
C
C
′
concur.
9
1
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Reflection of incenter across midpoint of AB
Let
T
1
T_1
T
1
and
T
2
T_2
T
2
be the points of tangency of the excircles of a triangle
A
B
C
ABC
A
BC
with its sides
B
C
BC
BC
and
A
C
AC
A
C
respectively. It is known that the reflection of the incenter of
A
B
C
ABC
A
BC
across the midpoint of
A
B
AB
A
B
lies on the circumcircle of triangle
C
T
1
T
2
CT_1T_2
C
T
1
T
2
. Find
∠
B
C
A
\angle BCA
∠
BC
A
.
8
4
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7
4
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6
4
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5
4
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3
4
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2
4
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1
4
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16
1
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Show that CK is parallel to AB
The incircle of triangle
A
B
C
ABC
A
BC
touches
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at points
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
, respectively. The perpendicular from the incenter
I
I
I
to the median from vertex
C
C
C
meets the line
A
1
B
1
A_1B_1
A
1
B
1
in point
K
K
K
. Prove that
C
K
CK
C
K
is parallel to
A
B
AB
A
B
.
4
4
Show problems
21
1
Hide problems
Show that (DEN) passes through the midpoint of BC
Chords
B
C
BC
BC
and
D
E
DE
D
E
of circle
ω
\omega
ω
meet at point
A
A
A
. The line through
D
D
D
parallel to
B
C
BC
BC
meets
ω
\omega
ω
again at
F
F
F
, and
F
A
FA
F
A
meets
ω
\omega
ω
again at
T
T
T
. Let
M
=
E
T
∩
B
C
M = ET \cap BC
M
=
ET
∩
BC
and let
N
N
N
be the reflection of
A
A
A
over
M
M
M
. Show that
(
D
E
N
)
(DEN)
(
D
EN
)
passes through the midpoint of
B
C
BC
BC
.
15
1
Hide problems
Prove that two inscribed triangles are congruent
(a) 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 inscribed into triangle
A
B
C
ABC
A
BC
so that
C
1
A
1
⊥
B
C
C_1A_1 \perp BC
C
1
A
1
⊥
BC
,
A
1
B
1
⊥
C
A
A_1B_1 \perp CA
A
1
B
1
⊥
C
A
,
B
1
C
1
⊥
A
B
B_1C_1 \perp AB
B
1
C
1
⊥
A
B
,
B
2
A
2
⊥
B
C
B_2A_2 \perp BC
B
2
A
2
⊥
BC
,
C
2
B
2
⊥
C
A
C_2B_2 \perp CA
C
2
B
2
⊥
C
A
,
A
2
C
2
⊥
A
B
A_2C_2 \perp AB
A
2
C
2
⊥
A
B
. Prove that these triangles are equal.(b) Points
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
,
A
2
A_2
A
2
,
B
2
B_2
B
2
,
C
2
C_2
C
2
lie inside a triangle
A
B
C
ABC
A
BC
so that
A
1
A_1
A
1
is on segment
A
B
1
AB_1
A
B
1
,
B
1
B_1
B
1
is on segment
B
C
1
BC_1
B
C
1
,
C
1
C_1
C
1
is on segment
C
A
1
CA_1
C
A
1
,
A
2
A_2
A
2
is on segment
A
C
2
AC_2
A
C
2
,
B
2
B_2
B
2
is on segment
B
A
2
BA_2
B
A
2
,
C
2
C_2
C
2
is on segment
C
B
2
CB_2
C
B
2
, and the angles
B
A
A
1
BAA_1
B
A
A
1
,
C
B
B
2
CBB_2
CB
B
2
,
A
C
C
1
ACC_1
A
C
C
1
,
C
A
A
2
CAA_2
C
A
A
2
,
A
B
B
2
ABB_2
A
B
B
2
,
B
C
C
2
BCC_2
BC
C
2
are equal. 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.
14
1
Hide problems
Right-angled trapezoid ABCD
Let
M
M
M
,
N
N
N
be the midpoints of diagonals
A
C
AC
A
C
,
B
D
BD
B
D
of a right-angled trapezoid
A
B
C
D
ABCD
A
BC
D
(
∡
A
=
∡
D
=
9
0
∘
\measuredangle A=\measuredangle D = 90^\circ
∡
A
=
∡
D
=
9
0
∘
). The circumcircles of triangles
A
B
N
ABN
A
BN
,
C
D
M
CDM
C
D
M
meet the line
B
C
BC
BC
in the points
Q
Q
Q
,
R
R
R
. Prove that the distances from
Q
Q
Q
,
R
R
R
to the midpoint of
M
N
MN
MN
are equal.