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Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2017 Sharygin Geometry Olympiad
2017 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(32)
P24
1
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similarities with 2 tetrahedrons
Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?
P23
1
Hide problems
peprendicular from I to AI,BI,CI lead to concurrent
Let a line
m
m
m
touch the incircle of triangle
A
B
C
ABC
A
BC
. The lines passing through the incenter
I
I
I
and perpendicular to
A
I
,
B
I
,
C
I
AI, BI, CI
A
I
,
B
I
,
C
I
meet
m
m
m
at points
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
respectively. Prove that
A
A
′
,
B
B
′
AA', BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
concur.
P22
1
Hide problems
distance from a point on diagonal of a cyclic quadrilateral
Let
P
P
P
be an arbitrary point on the diagonal
A
C
AC
A
C
of cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
, and
P
K
,
P
L
,
P
M
,
P
N
,
P
O
PK, PL, PM, PN, PO
P
K
,
P
L
,
PM
,
PN
,
PO
be the perpendiculars from
P
P
P
to
A
B
,
B
C
,
C
D
,
D
A
,
B
D
AB, BC, CD, DA, BD
A
B
,
BC
,
C
D
,
D
A
,
B
D
respectively. Prove that the distance from
P
P
P
to
K
N
KN
K
N
is equal to the distance from
O
O
O
to
M
L
ML
M
L
.
P21
1
Hide problems
joining midpoints of opposite sides of circumscribed hexagon
A convex hexagon is circumscribed about a circle of radius
1
1
1
. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number
r
r
r
such that the length of at least one segment is at least
r
.
r.
r
.
P20
1
Hide problems
locus with reflections on 2 perpendicular lines
Given a right-angled triangle
A
B
C
ABC
A
BC
and two perpendicular lines
x
x
x
and
y
y
y
passing through the vertex
A
A
A
of its right angle. For an arbitrary point
X
X
X
on
x
x
x
define
y
B
y_B
y
B
and
y
C
y_C
y
C
as the reflections of
y
y
y
about
X
B
XB
XB
and
X
C
XC
XC
respectively. Let
Y
Y
Y
be the common point of
y
b
y_b
y
b
and
y
c
y_c
y
c
. Find the locus of
Y
Y
Y
(when
y
b
y_b
y
b
and
y
c
y_c
y
c
do not coincide).
P19
1
Hide problems
circumcircles - cevians - midpoints, concurrency wanted
Let cevians
A
A
′
,
B
B
′
AA', BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
of triangle
A
B
C
ABC
A
BC
concur at point
P
.
P.
P
.
The circumcircle of triangle
P
A
′
B
′
PA'B'
P
A
′
B
′
meets
A
C
AC
A
C
and
B
C
BC
BC
at points
M
M
M
and
N
N
N
respectively, and the circumcircles of triangles
P
C
′
B
′
PC'B'
P
C
′
B
′
and
P
A
′
C
′
PA'C'
P
A
′
C
′
meet
A
C
AC
A
C
and
B
C
BC
BC
for the second time respectively at points
K
K
K
and
L
L
L
. The line
c
c
c
passes through the midpoints of segments
M
N
MN
MN
and
K
L
KL
K
L
. The lines
a
a
a
and
b
b
b
are defined similarly. Prove that
a
a
a
,
b
b
b
and
c
c
c
concur.
P18
1
Hide problems
angle relation with symmedian point and feet of altitude
Let
L
L
L
be the common point of the symmedians of triangle
A
B
C
ABC
A
BC
, and
B
H
BH
B
H
be its altitude. It is known that
∠
A
L
H
=
18
0
o
−
2
∠
A
\angle ALH = 180^o -2\angle A
∠
A
L
H
=
18
0
o
−
2∠
A
. Prove that
∠
C
L
H
=
18
0
o
−
2
∠
C
\angle CLH = 180^o - 2\angle C
∠
C
L
H
=
18
0
o
−
2∠
C
.
P17
1
Hide problems
construct K in acut ABC with <KBA = 2<KAB, <KBC = 2<KCB
Using a compass and a ruler, construct a point
K
K
K
inside an acute-angled triangle
A
B
C
ABC
A
BC
so that
∠
K
B
A
=
2
∠
K
A
B
\angle KBA = 2\angle KAB
∠
K
B
A
=
2∠
K
A
B
and
∠
K
B
C
=
2
∠
K
C
B
\angle KBC = 2\angle KCB
∠
K
BC
=
2∠
K
CB
.
P16
1
Hide problems
points definded by projections of intersections of tangents on sides
The tangents to the circumcircle of triangle
A
B
C
ABC
A
BC
at
A
A
A
and
B
B
B
meet at point
D
D
D
. The circle passing through the projections of
D
D
D
to
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
, meet
A
B
AB
A
B
for the second time at point
C
′
C'
C
′
. Points
A
′
,
B
′
A', B'
A
′
,
B
′
are defined similarly. Prove that
A
A
′
,
B
B
′
,
C
C
′
AA', BB', CC'
A
A
′
,
B
B
′
,
C
C
′
concur.
P15
1
Hide problems
two circumscribed quadrilateral relates to touchpoints of incircle
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with incircle
ω
\omega
ω
and incenter
I
I
I
. Let
ω
\omega
ω
touch
A
B
,
B
C
AB, BC
A
B
,
BC
and
C
A
CA
C
A
at points
D
,
E
,
F
D, E, F
D
,
E
,
F
respectively. The circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
centered at
J
1
J_1
J
1
and
J
2
J_2
J
2
respectively are inscribed into A
D
I
F
DIF
D
I
F
and
B
D
I
E
BDIE
B
D
I
E
. Let
J
1
J
2
J_1J_2
J
1
J
2
intersect
A
B
AB
A
B
at point
M
M
M
. Prove that
C
D
CD
C
D
is perpendicular to
I
M
IM
I
M
.
P5
1
Hide problems
locus of incenters & vertices of triangles with given side
A segment
A
B
AB
A
B
is fixed on the plane. Consider all acute-angled triangles with side
A
B
AB
A
B
. Find the locus of а) the vertices of their greatest angles, b) their incenters.
8
3
Hide problems
Median through point on circumcircle of square
Let
A
B
C
D
ABCD
A
BC
D
be a square, and let
P
P
P
be a point on the minor arc
C
D
CD
C
D
of its circumcircle. The lines
P
A
,
P
B
PA, PB
P
A
,
PB
meet the diagonals
B
D
,
A
C
BD, AC
B
D
,
A
C
at points
K
,
L
K, L
K
,
L
respectively. The points
M
,
N
M, N
M
,
N
are the projections of
K
,
L
K, L
K
,
L
respectively to
C
D
CD
C
D
, and
Q
Q
Q
is the common point of lines
K
N
KN
K
N
and
M
L
ML
M
L
. Prove that
P
Q
PQ
PQ
bisects the segment
A
B
AB
A
B
.
Isotomic points intercepted by common inner tangents
Let
A
K
AK
A
K
and
B
L
BL
B
L
be the altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
, and let
ω
\omega
ω
be the excircle of
A
B
C
ABC
A
BC
touching side
A
B
AB
A
B
. The common internal tangents to circles
C
K
L
CKL
C
K
L
and
ω
\omega
ω
meet
A
B
AB
A
B
at points
P
P
P
and
Q
Q
Q
. Prove that
A
P
=
B
Q
AP =BQ
A
P
=
BQ
.Proposed by I.Frolov
Special partition of set.
10.8 Suppose
S
S
S
is a set of points in the plane,
∣
S
∣
|S|
∣
S
∣
is even; no three points of
S
S
S
are collinear. Prove that
S
S
S
can be partitioned into two sets
S
1
S_1
S
1
and
S
2
S_2
S
2
so that their convex hulls have equal number of vertices.
7
3
Hide problems
Arbitrary lines made by regular 13-gons concur
Let
A
1
A
2
…
A
13
A_1A_2 \dots A_{13}
A
1
A
2
…
A
13
and
B
1
B
2
…
B
13
B_1B_2 \dots B_{13}
B
1
B
2
…
B
13
be two regular
13
13
13
-gons in the plane such that the points
B
1
B_1
B
1
and
A
13
A_{13}
A
13
coincide and lie on the segment
A
1
B
13
A_1B_{13}
A
1
B
13
, and both polygons lie in the same semiplane with respect to this segment. Prove that the lines
A
1
A
9
,
B
13
B
8
A_1A_9, B_{13}B_8
A
1
A
9
,
B
13
B
8
and
A
8
B
9
A_8B_9
A
8
B
9
are concurrent.
Maximise number of acute triangles
Let
a
a
a
and
b
b
b
be parallel lines with
50
50
50
distinct points marked on
a
a
a
and
50
50
50
distinct points marked on
b
b
b
. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.
Orthogonal circles in bicentric polygon
10.7 A quadrilateral
A
B
C
D
ABCD
A
BC
D
is circumscribed around the circle
ω
\omega
ω
centered at
I
I
I
and inscribed into the circle
Γ
\Gamma
Γ
. The lines
A
B
,
C
D
AB, CD
A
B
,
C
D
meet at point
P
P
P
, and the lines
B
C
,
A
D
BC, AD
BC
,
A
D
meet at point
Q
Q
Q
. Prove that the circles
⊙
(
P
I
Q
)
\odot(PIQ)
⊙
(
P
I
Q
)
and
Γ
\Gamma
Γ
are orthogonal.
6
3
Hide problems
Covering bisected triangles with semi-circumdisc
A median of an acute-angled triangle dissects it into two triangles. Prove that each of them can be covered by a semidisc congruent to a half of the circumdisc of the initial triangle.
Ratio of areas erected on right triangle
Let
A
B
C
ABC
A
BC
be a right-angled triangle (
∠
C
=
9
0
∘
\angle C = 90^\circ
∠
C
=
9
0
∘
) and
D
D
D
be the midpoint of an altitude from C. The reflections of the line
A
B
AB
A
B
about
A
D
AD
A
D
and
B
D
BD
B
D
, respectively, meet at point
F
F
F
. Find the ratio
S
A
B
F
:
S
A
B
C
S_{ABF}:S_{ABC}
S
A
BF
:
S
A
BC
. Note:
S
α
S_{\alpha}
S
α
means the area of
α
\alpha
α
.
Inspheres and sum of angles
10.6 Let the insphere of a pyramid
S
A
B
C
SABC
S
A
BC
touch the faces
S
A
B
,
S
B
C
,
S
C
A
SAB, SBC, SCA
S
A
B
,
SBC
,
SC
A
at
D
,
E
,
F
D, E, F
D
,
E
,
F
respectively. Find all the possible values of the sum of the angles
S
D
A
,
S
E
B
,
S
F
C
SDA, SEB, SFC
S
D
A
,
SEB
,
SFC
.
5
3
Hide problems
Inner tangent passes through midpoint
A square
A
B
C
D
ABCD
A
BC
D
is given. Two circles are inscribed into angles
A
A
A
and
B
B
B
, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of
A
B
AB
A
B
.
Line parallel to base
Let
B
H
b
,
C
H
c
BH_b, CH_c
B
H
b
,
C
H
c
be altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
. The line
H
b
H
c
H_bH_c
H
b
H
c
meets the circumcircle of
A
B
C
ABC
A
BC
at points
X
X
X
and
Y
Y
Y
. Points
P
,
Q
P,Q
P
,
Q
are the reflections of
X
,
Y
X,Y
X
,
Y
about
A
B
,
A
C
AB,AC
A
B
,
A
C
respectively. Prove that
P
Q
∥
B
C
PQ \parallel BC
PQ
∥
BC
.Proposed by Pavel Kozhevnikov
Concyclic points and miquel point
10.5 Let
B
B
′
BB'
B
B
′
,
C
C
′
CC'
C
C
′
be the altitudes of an acute triangle
A
B
C
ABC
A
BC
. Two circles through
A
A
A
and
C
′
C'
C
′
are tangent to
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
. Prove that
A
,
B
′
,
P
,
Q
A, B', P, Q
A
,
B
′
,
P
,
Q
are concyclic.
1
3
Hide problems
Parallel line in cyclic kite
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with
A
B
=
B
C
AB=BC
A
B
=
BC
and
A
D
=
C
D
AD = CD
A
D
=
C
D
. A point
M
M
M
lies on the minor arc
C
D
CD
C
D
of its circumcircle. The lines
B
M
BM
BM
and
C
D
CD
C
D
meet at point
P
P
P
, the lines
A
M
AM
A
M
and
B
D
BD
B
D
meet at point
Q
Q
Q
. Prove that
P
Q
∥
A
C
PQ \parallel AC
PQ
∥
A
C
.
Equilateral triangle with ratio of regular pentagon
Let
A
B
C
ABC
A
BC
be a regular triangle. The line passing through the midpoint of
A
B
AB
A
B
and parallel to
A
C
AC
A
C
meets the minor arc
A
B
AB
A
B
of the circumcircle at point
K
K
K
. Prove that the ratio
A
K
:
B
K
AK:BK
A
K
:
B
K
is equal to the ratio of the side and the diagonal of a regular pentagon.
Midpoint of two circumcentre
If two circles intersect at
A
,
B
A,B
A
,
B
and common tangents of them intesrsect circles at
C
,
D
C,D
C
,
D
if
O
a
O_a
O
a
is circumcentre of
A
C
D
ACD
A
C
D
and
O
b
O_b
O
b
is circumcentre of
B
C
D
BCD
BC
D
prove
A
B
AB
A
B
intersects
O
a
O
b
O_aO_b
O
a
O
b
at its midpoint
2
3
Hide problems
Angle bisector through circumcenter
Let
H
H
H
and
O
O
O
be the orthocenter and circumcenter of an acute-angled triangle
A
B
C
ABC
A
BC
, respectively. The perpendicular bisector of
B
H
BH
B
H
meets
A
B
AB
A
B
and
B
C
BC
BC
at points
A
1
A_1
A
1
and
C
1
C_1
C
1
, respectively. Prove that
O
B
OB
OB
bisects the angle
A
1
O
C
1
A_1OC_1
A
1
O
C
1
.
Mixtilinear touchpoint is midpoint of side
Let
I
I
I
be the incenter of a triangle
A
B
C
ABC
A
BC
,
M
M
M
be the midpoint of
A
C
AC
A
C
, and
W
W
W
be the midpoint of arc
A
B
AB
A
B
of the circumcircle not containing
C
C
C
. It is known that
∠
A
I
M
=
9
0
∘
\angle AIM = 90^\circ
∠
A
I
M
=
9
0
∘
. Find the ratio
C
I
:
I
W
CI:IW
C
I
:
I
W
.
Easy inequality in ABC
If
A
B
C
ABC
A
BC
is acute triangle, prove distance from each vertex to corresponding excentre is less than sum of two greatest side of triangle
3
3
Hide problems
Concentric circles in centroid configuration
Let
A
D
,
B
E
AD, BE
A
D
,
BE
and
C
F
CF
CF
be the medians of triangle
A
B
C
ABC
A
BC
. The points
X
X
X
and
Y
Y
Y
are the reflections of
F
F
F
about
A
D
AD
A
D
and
B
E
BE
BE
, respectively. Prove that the circumcircles of triangles
B
E
X
BEX
BEX
and
A
D
Y
ADY
A
D
Y
are concentric.
Obtuse angle at variable midpoint
The angles
B
B
B
and
C
C
C
of an acute-angled triangle
A
B
C
ABC
A
BC
are greater than
6
0
∘
60^\circ
6
0
∘
. Points
P
,
Q
P,Q
P
,
Q
are chosen on the sides
A
B
,
A
C
AB,AC
A
B
,
A
C
respectively so that the points
A
,
P
,
Q
A,P,Q
A
,
P
,
Q
are concyclic with the orthocenter
H
H
H
of the triangle
A
B
C
ABC
A
BC
. Point
K
K
K
is the midpoint of
P
Q
PQ
PQ
. Prove that
∠
B
K
C
>
9
0
∘
\angle BKC > 90^\circ
∠
B
K
C
>
9
0
∘
. Proposed by A. Mudgal
Power×area
A
B
C
D
ABCD
A
BC
D
is convex quadrilateral. If
W
a
W_a
W
a
is product of power of
A
A
A
about circle
B
C
D
BCD
BC
D
and area of triangle
B
C
D
BCD
BC
D
. And define
W
b
,
W
c
,
W
d
W_b,W_c,W_d
W
b
,
W
c
,
W
d
similarly.prove
W
a
+
W
b
+
W
c
+
W
d
=
0
W_a+W_b+W_c+W_d=0
W
a
+
W
b
+
W
c
+
W
d
=
0
4
3
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Congruent triangles after several cuts
Alex dissects a paper triangle into two triangles. Each minute after this he dissects one of obtained triangles into two triangles. After some time (at least one hour) it appeared that all obtained triangles were congruent. Find all initial triangles for which this is possible.
Moving circles touch in isosceles triangle
Points
M
M
M
and
K
K
K
are chosen on lateral sides
A
B
,
A
C
AB,AC
A
B
,
A
C
of an isosceles triangle
A
B
C
ABC
A
BC
and point
D
D
D
is chosen on
B
C
BC
BC
such that
A
M
D
K
AMDK
A
M
DK
is a parallelogram. Let the lines
M
K
MK
M
K
and
B
C
BC
BC
meet at point
L
L
L
, and let
X
,
Y
X,Y
X
,
Y
be the intersection points of
A
B
,
A
C
AB,AC
A
B
,
A
C
with the perpendicular line from
D
D
D
to
B
C
BC
BC
. Prove that the circle with center
L
L
L
and radius
L
D
LD
L
D
and the circumcircle of triangle
A
X
Y
AXY
A
X
Y
are tangent.
8 lines and incircle
Given triangle
A
B
C
ABC
A
BC
and its incircle
ω
\omega
ω
prove you can use just a ruler and drawing at most 8 lines to construct points
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
on
ω
\omega
ω
such that
A
,
B
′
,
C
′
A,B',C'
A
,
B
′
,
C
′
and
B
,
C
′
,
A
′
B,C',A'
B
,
C
′
,
A
′
and
C
,
A
′
,
B
′
C,A',B'
C
,
A
′
,
B
′
are collinear.
P14
1
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Three circles
Let points
B
B
B
and
C
C
C
lie on the circle with diameter
A
D
AD
A
D
and center
O
O
O
on the same side of
A
D
AD
A
D
. The circumcircles of triangles
A
B
O
ABO
A
BO
and
C
D
O
CDO
C
D
O
meet
B
C
BC
BC
at points
F
F
F
and
E
E
E
respectively. Prove that
R
2
=
A
F
.
D
E
R^2 = AF.DE
R
2
=
A
F
.
D
E
, where
R
R
R
is the radius of the given circle. Proposed by N.Moskvitin
P13
1
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Tangents parallel to tangents
Two circles pass through points
A
A
A
and
B
B
B
. A third circle touches both these circles and meets
A
B
AB
A
B
at points
C
C
C
and
D
D
D
. Prove that the tangents to this circle at these points are parallel to the common tangents of two given circles. Proposed by A.Zaslavsky
P12
1
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Symmedian passes through intersection of two circumcircle
Let
A
A
1
,
C
C
1
AA_1 , CC_1
A
A
1
,
C
C
1
be the altitudes of triangle
A
B
C
,
B
0
ABC, B_0
A
BC
,
B
0
the common point of the altitude from
B
B
B
and the circumcircle of
A
B
C
ABC
A
BC
; and
Q
Q
Q
the common point of the circumcircles of
A
B
C
ABC
A
BC
and
A
1
C
1
B
0
A_1C_1B_0
A
1
C
1
B
0
, distinct from
B
0
B_0
B
0
. Prove that
B
Q
BQ
BQ
is the symmedian of
A
B
C
ABC
A
BC
. Proposed by D.Shvetsov
P11
1
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Set with circumcenters of each triangle in set
A finite number of points is marked on the plane. Each three of them are not collinear. A circle is circumscribed around each triangle with marked vertices. Is it possible that all centers of these circles are also marked? Proposed by A.Tolesnikov
P10
1
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Circumcenter equidistant from two points
Points
K
K
K
and
L
L
L
on the sides
A
B
AB
A
B
and
B
C
BC
BC
of parallelogram
A
B
C
D
ABCD
A
BC
D
are such that
∠
A
K
D
=
∠
C
L
D
\angle AKD = \angle CLD
∠
A
KD
=
∠
C
L
D
. Prove that the circumcenter of triangle
B
K
L
BKL
B
K
L
is equidistant from
A
A
A
and
C
C
C
. Proposed by I.I.Bogdanov
P9
1
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2 lines meet on the altitude
Let
C
0
C_0
C
0
be the midpoint of hypotenuse
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
;
A
A
1
,
B
B
1
AA_1, BB_1
A
A
1
,
B
B
1
the bisectors of this triangle;
I
I
I
its incenter. Prove that the lines
C
0
I
C_0I
C
0
I
and
A
1
B
1
A_1B_1
A
1
B
1
meet on the altitude from
C
C
C
. Proposed by A.Zaslavsky
P8
1
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Circumcenter of triangle on diagonal
Let
A
D
AD
A
D
be the base of trapezoid
A
B
C
D
ABCD
A
BC
D
. It is known that the circumcenter of triangle
A
B
C
ABC
A
BC
lies on
B
D
BD
B
D
. Prove that the circumcenter of triangle
A
B
D
ABD
A
B
D
lies on
A
C
AC
A
C
. Proposed by Ye.Bakayev
P7
1
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Ratio of sides less than 2
The circumcenter of a triangle lies on its incircle. Prove that the ratio of its greatest and smallest sides is less than two. Proposed by B.Frenkin
P6
1
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Line passes through the touching points of incircle
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
A
C
=
B
D
=
A
D
AC = BD = AD
A
C
=
B
D
=
A
D
;
E
E
E
and
F
F
F
the midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
respectively;
O
O
O
the common point of the diagonals.Prove that
E
F
EF
EF
passes through the touching points of the incircle of triangle
A
O
D
AOD
A
O
D
with
A
O
AO
A
O
and
O
D
OD
O
D
Proposed by N.Moskvitin
P4
1
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To find angles of the triangle
A triangle
A
B
C
ABC
A
BC
is given. Let C\ensuremath{'} be the vertex of an isosceles triangle ABC\ensuremath{'} with \angle C\ensuremath{'} = 120^{\circ} constructed on the other side of
A
B
AB
A
B
than
C
C
C
, and B\ensuremath{'} be the vertex of an equilateral triangle ACB\ensuremath{'} constructed on the same side of
A
C
AC
A
C
as
A
B
C
ABC
A
BC
. Let
K
K
K
be the midpoint of BB\ensuremath{'} Find the angles of triangle KCC\ensuremath{'}. Proposed by A.Zaslavsky
P3
1
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Orthocenter and tangency point are collinear
Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
;
H
B
,
H
C
H_B, H_C
H
B
,
H
C
the orthocenters of triangles
A
C
I
ACI
A
C
I
and
A
B
I
ABI
A
B
I
respectively;
K
K
K
the touching point of the incircle with the side
B
C
BC
BC
. Prove that
H
B
,
H
C
H_B, H_C
H
B
,
H
C
and K are collinear. Proposed by M.Plotnikov
P2
1
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Diagonals of quadriateral are equal
A circle cuts off four right-angled triangles from rectangle
A
B
C
D
ABCD
A
BC
D
.Let
A
0
,
B
0
,
C
0
A_0, B_0, C_0
A
0
,
B
0
,
C
0
and
D
0
D_0
D
0
be the midpoints of the correspondent hypotenuses. Prove that
A
0
C
0
=
B
0
D
0
A_0C_0 = B_0D_0
A
0
C
0
=
B
0
D
0
Proosed by L.Shteingarts
P1
1
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Prime quadrilateral
Mark on a cellular paper four nodes forming a convex quadrilateral with the sidelengths equal to four different primes. (Proposed by A.Zaslavsky)