MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2019 Sharygin Geometry Olympiad
2019 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(24)
11
1
Hide problems
Sharygin CR 2019 P11
Morteza marks six points in the plane. He then calculates and writes down the area of every triangle with vertices in these points (
20
20
20
numbers). Is it possible that all of these numbers are integers, and that they add up to
2019
2019
2019
?
24
1
Hide problems
Sharygin CR 2019 P24
Two unit cubes have a common center. Is it always possible to number the vertices of each cube from
1
1
1
to
8
8
8
so that the distance between each pair of identically numbered vertices would be at most
4
/
5
4/5
4/5
? What about at most
13
/
16
13/16
13/16
?
23
1
Hide problems
Sharygin CR 2019 P23
In the plane, let
a
a
a
,
b
b
b
be two closed broken lines (possibly self-intersecting), and
K
K
K
,
L
L
L
,
M
M
M
,
N
N
N
be four points. The vertices of
a
a
a
,
b
b
b
and the points
K
K
K
L
L
L
,
M
M
M
,
N
N
N
are in general position (i.e. no three of these points are collinear, and no three segments between them concur at an interior point). Each of segments
K
L
KL
K
L
and
M
N
MN
MN
meets
a
a
a
at an even number of points, and each of segments
L
M
LM
L
M
and
N
K
NK
N
K
meets
a
a
a
at an odd number of points. Conversely, each of segments
K
L
KL
K
L
and
M
N
MN
MN
meets
b
b
b
at an odd number of points, and each of segments
L
M
LM
L
M
and
N
K
NK
N
K
meets
b
b
b
at an even number of points. Prove that
a
a
a
and
b
b
b
intersect.
21
1
Hide problems
Sharygin CR 2019 P21
An ellipse
Γ
\Gamma
Γ
and its chord
A
B
AB
A
B
are given. Find the locus of orthocenters of triangles
A
B
C
ABC
A
BC
inscribed into
Γ
\Gamma
Γ
.
10
1
Hide problems
Sharygin CR 2019 P10
Let
N
N
N
be the midpoint of arc
A
B
C
ABC
A
BC
of the circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
, and
N
P
NP
NP
,
N
T
NT
NT
be the tangents to the incircle of this triangle. The lines
B
P
BP
BP
and
B
T
BT
BT
meet the circumcircle for the second time at points
P
1
P_1
P
1
and
T
1
T_1
T
1
respectively. Prove that
P
P
1
=
T
T
1
PP_1 = TT_1
P
P
1
=
T
T
1
.
20
1
Hide problems
Sharygin CR 2019 P20
Let
O
O
O
be the circumcenter of triangle ABC,
H
H
H
be its orthocenter, and
M
M
M
be the midpoint of
A
B
AB
A
B
. The line
M
H
MH
M
H
meets the line passing through
O
O
O
and parallel to
A
B
AB
A
B
at point
K
K
K
lying on the circumcircle of
A
B
C
ABC
A
BC
. Let
P
P
P
be the projection of
K
K
K
onto
A
C
AC
A
C
. Prove that
P
H
∥
B
C
PH \parallel BC
P
H
∥
BC
.
19
1
Hide problems
Sharygin CR 2019 P19
Let
A
L
a
AL_a
A
L
a
,
B
L
b
BL_b
B
L
b
,
C
L
c
CL_c
C
L
c
be the bisecors of triangle
A
B
C
ABC
A
BC
. The tangents to the circumcircle of
A
B
C
ABC
A
BC
at
B
B
B
and
C
C
C
meet at point
K
a
K_a
K
a
, points
K
b
K_b
K
b
,
K
c
K_c
K
c
are defined similarly. Prove that the lines
K
a
L
a
K_aL_a
K
a
L
a
,
K
b
L
b
K_bL_b
K
b
L
b
and
K
c
L
c
K_cL_c
K
c
L
c
concur.
9
1
Hide problems
Sharygin CR 2019 P9
Let
A
M
A_M
A
M
be the midpoint of side
B
C
BC
BC
of an acute-angled
Δ
A
B
C
\Delta ABC
Δ
A
BC
, and
A
H
A_H
A
H
be the foot of the altitude to this side. Points
B
M
,
B
H
,
C
M
,
C
H
B_M, B_H, C_M, C_H
B
M
,
B
H
,
C
M
,
C
H
are defined similarly. Prove that one of the ratios
A
M
A
H
:
A
H
A
,
B
M
B
H
:
B
H
B
,
C
M
C
H
:
C
H
C
A_MA_H : A_HA, B_MB_H : B_HB, C_MC_H : C_HC
A
M
A
H
:
A
H
A
,
B
M
B
H
:
B
H
B
,
C
M
C
H
:
C
H
C
is equal to the sum of two remaining ratios
18
1
Hide problems
Sharygin CR 2019 P18
A quadrilateral
A
B
C
D
ABCD
A
BC
D
without parallel sidelines is circumscribed around a circle centered at
I
I
I
. Let
K
,
L
,
M
K, L, M
K
,
L
,
M
and
N
N
N
be the midpoints of
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
respectively. It is known that
A
B
⋅
C
D
=
4
I
K
⋅
I
M
AB \cdot CD = 4IK \cdot IM
A
B
⋅
C
D
=
4
I
K
⋅
I
M
. Prove that
B
C
⋅
A
D
=
4
I
L
⋅
I
N
BC \cdot AD = 4IL \cdot IN
BC
⋅
A
D
=
4
I
L
⋅
I
N
.
17
1
Hide problems
Sharygin CR 2019 P17
Three circles
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
,
ω
3
\omega_3
ω
3
are given. Let
A
0
A_0
A
0
and
A
1
A_1
A
1
be the common points of
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
,
B
0
B_0
B
0
and
B
1
B_1
B
1
be the common points of
ω
2
\omega_2
ω
2
and
ω
3
\omega_3
ω
3
,
C
0
C_0
C
0
and
C
1
C_1
C
1
be the common points of
ω
3
\omega_3
ω
3
and
ω
1
\omega_1
ω
1
. Let
O
i
,
j
,
k
O_{i,j,k}
O
i
,
j
,
k
be the circumcenter of triangle
A
i
B
j
C
k
A_iB_jC_k
A
i
B
j
C
k
. Prove that the four lines of the form
O
i
j
k
O
1
−
i
,
1
−
j
,
1
−
k
O_{ijk}O_{1 - i,1 - j,1 - k}
O
ijk
O
1
−
i
,
1
−
j
,
1
−
k
are concurrent or parallel.
8
4
Show problems
16
1
Hide problems
Sharygin CR 2019 P16
Let
A
H
1
AH_1
A
H
1
and
B
H
2
BH_2
B
H
2
be the altitudes of triangle
A
B
C
ABC
A
BC
. Let the tangent to the circumcircle of
A
B
C
ABC
A
BC
at
A
A
A
meet
B
C
BC
BC
at point
S
1
S_1
S
1
, and the tangent at
B
B
B
meet
A
C
AC
A
C
at point
S
2
S_2
S
2
. Let
T
1
T_1
T
1
and
T
2
T_2
T
2
be the midpoints of
A
S
1
AS_1
A
S
1
and
B
S
2
BS_2
B
S
2
respectively. Prove that
T
1
T
2
T_1T_2
T
1
T
2
,
A
B
AB
A
B
and
H
1
H
2
H_1H_2
H
1
H
2
concur.
15
1
Hide problems
Sharygin CR 2019 P15
The incircle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
touches the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
at points
D
D
D
,
E
E
E
and
F
F
F
respectively. The perpendicular from
E
E
E
to
D
F
DF
D
F
meets
B
C
BC
BC
at point
X
X
X
, and the perpendicular from
F
F
F
to
D
E
DE
D
E
meets
B
C
BC
BC
at point
Y
Y
Y
. The segment
A
D
AD
A
D
meets
ω
\omega
ω
for the second time at point
Z
Z
Z
. Prove that the circumcircle of the triangle
X
Y
Z
XYZ
X
Y
Z
touches
ω
\omega
ω
.
7
3
Hide problems
Sharygin CR 2019 P7
Let
A
H
A
AH_A
A
H
A
,
B
H
B
BH_B
B
H
B
,
C
H
C
CH_C
C
H
C
be the altitudes of the acute-angled
Δ
A
B
C
\Delta ABC
Δ
A
BC
. Let
X
X
X
be an arbitrary point of segment
C
H
C
CH_C
C
H
C
, and
P
P
P
be the common point of circles with diameters
H
C
X
H_CX
H
C
X
and BC, distinct from
H
C
H_C
H
C
. The lines
C
P
CP
CP
and
A
H
A
AH_A
A
H
A
meet at point
Q
Q
Q
, and the lines
X
P
XP
XP
and
A
B
AB
A
B
meet at point
R
R
R
. Prove that
A
,
P
,
Q
,
R
,
H
B
A, P, Q, R, H_B
A
,
P
,
Q
,
R
,
H
B
are concyclic.
Medial line of a triangle
Let the incircle
ω
\omega
ω
of
△
A
B
C
\triangle ABC
△
A
BC
touch
A
C
AC
A
C
and
A
B
AB
A
B
at points
E
E
E
and
F
F
F
respectively. Points
X
X
X
,
Y
Y
Y
of
ω
\omega
ω
are such that
∠
B
X
C
=
∠
B
Y
C
=
9
0
∘
\angle BXC=\angle BYC=90^{\circ}
∠
BXC
=
∠
B
Y
C
=
9
0
∘
. Prove that
E
F
EF
EF
and
X
Y
XY
X
Y
meet on the medial line of
A
B
C
ABC
A
BC
.
locus, incenter, incircle, parallel related
Let
P
P
P
be an arbitrary point on side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
. Let
K
K
K
be the incenter of triangle
P
A
B
PAB
P
A
B
. Let the incircle of triangle
P
A
C
PAC
P
A
C
touch
B
C
BC
BC
at
F
F
F
. Point
G
G
G
on
C
K
CK
C
K
is such that
F
G
/
/
P
K
FG // PK
FG
//
P
K
. Find the locus of
G
G
G
.
14
1
Hide problems
Sharygin CR 2019 P14
Let the side
A
C
AC
A
C
of triangle
A
B
C
ABC
A
BC
touch the incircle and the corresponding excircle at points
K
K
K
and
L
L
L
respectively. Let
P
P
P
be the projection of the incenter onto the perpendicular bisector of
A
C
AC
A
C
. It is known that the tangents to the circumcircle of triangle
B
K
L
BKL
B
K
L
at
K
K
K
and
L
L
L
meet on the circumcircle of
A
B
C
ABC
A
BC
. Prove that the lines
A
B
AB
A
B
and
B
C
BC
BC
touch the circumcircle of triangle
P
K
L
PKL
P
K
L
.
22
1
Hide problems
Sharygin CR P22
Let
A
A
0
AA_0
A
A
0
be the altitude of the isosceles triangle
A
B
C
(
A
B
=
A
C
)
ABC~(AB = AC)
A
BC
(
A
B
=
A
C
)
. A circle
γ
\gamma
γ
centered at the midpoint of
A
A
0
AA_0
A
A
0
touches
A
B
AB
A
B
and
A
C
AC
A
C
. Let
X
X
X
be an arbitrary point of line
B
C
BC
BC
. Prove that the tangents from
X
X
X
to
γ
\gamma
γ
cut congruent segments on lines
A
B
AB
A
B
and
A
C
AC
A
C
13
1
Hide problems
Sharygin CR 2019 P13
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with altitude
A
T
=
h
AT = h
A
T
=
h
. The line passing through its circumcenter
O
O
O
and incenter
I
I
I
meets the sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
F
F
F
and
N
N
N
, respectively. It is known that
B
F
N
C
BFNC
BFNC
is a cyclic quadrilateral. Find the sum of the distances from the orthocenter of
A
B
C
ABC
A
BC
to its vertices.
6
4
Show problems
12
1
Hide problems
Max of min of cevian segments
Let
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
be an acute-angled triangle inscribed into a unit circle centered at
O
O
O
. The cevians from
A
i
A_i
A
i
passing through
O
O
O
meet the opposite sides at points
B
i
B_i
B
i
(
i
=
1
,
2
,
3
)
(i = 1, 2, 3)
(
i
=
1
,
2
,
3
)
respectively.[*] Find the minimal possible length of the longest of three segments
B
i
O
B_iO
B
i
O
. [*] Find the maximal possible length of the shortest of three segments
B
i
O
B_iO
B
i
O
.
5
4
Show problems
4
4
Show problems
3
4
Show problems
2
4
Show problems
1
4
Show problems