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National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2022 Sharygin Geometry Olympiad
2022 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(48)
8.5
1
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BD //AC , reflections of touchpoints of incircle wrt lines
An incircle of triangle
A
B
C
ABC
A
BC
touches
A
B
AB
A
B
,
B
C
BC
BC
,
A
C
AC
A
C
at points
C
1
C_1
C
1
,
A
1
A_1
A
1
,
B
1
B_1
B
1
respectively. Let
A
′
A'
A
′
be the reflection of
A
1
A_1
A
1
about
B
1
C
1
B_1C_1
B
1
C
1
, point
C
′
C'
C
′
is defined similarly. Lines
A
′
C
1
A'C_1
A
′
C
1
and
C
′
A
1
C'A_1
C
′
A
1
meet at point
D
D
D
. Prove that
B
D
∥
A
C
BD \parallel AC
B
D
∥
A
C
.
8.6
1
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ray construction, ratio OC/OD is maximal
Two circles meeting at points
A
,
B
A, B
A
,
B
and a point
O
O
O
lying outside them are given. Using a compass and a ruler construct a ray with origin
O
O
O
meeting the first circle at point
C
C
C
and the second one at point
D
D
D
in such a way that the ratio
O
C
:
O
D
OC : OD
OC
:
O
D
be maximal.
8.8
1
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inequality for daigonal on isosceles trapezoid
An isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
(
A
B
=
C
D
AB = CD
A
B
=
C
D
) is given. A point
P
P
P
on its circumcircle is such that segments
C
P
CP
CP
and
A
D
AD
A
D
meet at point
Q
Q
Q
. Let
L
L
L
be tha midpoint of
Q
D
QD
Q
D
. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line
P
L
PL
P
L
.
8.7
1
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10 points on a plane, no 4 od them on boundary of some square
Ten points on a plane a such that any four of them lie on the boundary of some square. Is obligatory true that all ten points lie on the boundary of some square?
8.4
1
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(AA_1B_1B) =(CC_1D_1D) , cyclic ABCD, 2 circles OPM, OPN
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral,
O
O
O
be its circumcenter,
P
P
P
be a common points of its diagonals, and
M
,
N
M , N
M
,
N
be the midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
respectively. A circle
O
P
M
OPM
OPM
meets for the second time segments
A
P
AP
A
P
and
B
P
BP
BP
at points
A
1
A_1
A
1
and
B
1
B_1
B
1
respectively and a circle
O
P
N
OPN
OPN
meets for the second time segments
C
P
CP
CP
and
D
P
DP
D
P
at points
C
1
C_1
C
1
and
D
1
D_1
D
1
respectively. Prove that the areas of quadrilaterals
A
A
1
B
1
B
AA_1B_1B
A
A
1
B
1
B
and
C
C
1
D
1
D
CC_1D_1D
C
C
1
D
1
D
are equal.
10.8
1
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centrosymmetric octahedron and Torricelli points,
Let
A
B
C
A
′
B
′
C
′
ABCA'B'C'
A
BC
A
′
B
′
C
′
be a centrosymmetric octahedron (vertices
A
A
A
and
A
′
A'
A
′
,
B
B
B
and
B
′
B'
B
′
,
C
C
C
and
C
′
C'
C
′
are opposite) such that the sums of four planar angles equal
24
0
o
240^o
24
0
o
for each vertex. The Torricelli points
T
1
T_1
T
1
and
T
2
T_2
T
2
of triangles
A
B
C
ABC
A
BC
and
A
′
B
C
A'BC
A
′
BC
are marked. Prove that the distances from
T
1
T_1
T
1
and
T
2
T_2
T
2
to
B
C
BC
BC
are equal.
10.7
1
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4 marked points inside each circle and 4 marked points outside each circle
Several circles are drawn on the plane and all points of their meeting or touching are marked. May be that each circle contains exactly four marked points and exactly four marked points lie on each circle?
10.6
1
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3 angle bicectors concur with OI
Let
O
,
I
O, I
O
,
I
be the circumcenter and the incenter of triangle
A
B
C
ABC
A
BC
,
P
P
P
be an arbitrary point on segment
O
I
OI
O
I
,
P
A
P_A
P
A
,
P
B
P_B
P
B
, and
P
C
P_C
P
C
be the second common points of lines
P
A
PA
P
A
,
P
B
PB
PB
, and
P
C
PC
PC
with the circumcircle of triangle
A
B
C
ABC
A
BC
. Prove that the bisectors of angles
B
P
A
C
BP_AC
B
P
A
C
,
C
P
B
A
CP_BA
C
P
B
A
, and
A
P
C
B
AP_CB
A
P
C
B
concur at a point lying on
O
I
OI
O
I
.
10.5
1
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common ext.tangents of MDP and MPE meet on midline of triangle ABC
Let
A
B
AB
A
B
and
A
C
AC
A
C
be the tangents from a point
A
A
A
to a circle
Ω
\Omega
Ω
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and
P
P
P
be an arbitrary point on this segment. A line
A
P
AP
A
P
meets
Ω
\Omega
Ω
at points
D
D
D
and
E
E
E
. Prove that the common external tangents to circles
M
D
P
MDP
M
D
P
and
M
P
E
MPE
MPE
meet on the midline of triangle
A
B
C
ABC
A
BC
.
10.3
1
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max no of points of a line, <AXC=1/2<BXC or <BXC =12<AXC
A line meets a segment
A
B
AB
A
B
at point
C
C
C
. Which is the maximal number of points
X
X
X
of this line such that one of angles
A
X
C
AXC
A
XC
and
B
X
C
BXC
BXC
is equlal to a half of the second one?
8.3
1
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locus of medicenter, equilateral triangle
A circle
ω
\omega
ω
and a point
P
P
P
not lying on it are given. Let
A
B
C
ABC
A
BC
be an arbitrary equilateral triangle inscribed into
ω
\omega
ω
and
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
be the projections of
P
P
P
to
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
. Find the locus of centroids of triangles
A
′
B
′
C
′
A' B'C'
A
′
B
′
C
′
.
8.2
1
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right-angled trapezoid, angle bisection
Let
A
B
C
D
ABCD
A
BC
D
be a right-angled trapezoid and
M
M
M
be the midpoint of its greater lateral side
C
D
CD
C
D
. Circumcircles
ω
1
\omega_{1}
ω
1
and
ω
2
\omega_{2}
ω
2
of triangles
B
C
M
BCM
BCM
and
A
M
D
AMD
A
M
D
meet for the second time at point
E
E
E
. Let
E
D
ED
E
D
meet
ω
1
\omega_{1}
ω
1
at point
F
F
F
, and
F
B
FB
FB
meet
A
D
AD
A
D
at point
G
G
G
. Prove that
G
M
GM
GM
bisects angle
B
G
D
BGD
BG
D
.
8.1
1
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Reflections of circumcenter
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
B
A
D
=
2
∠
B
C
D
\angle{BAD} = 2\angle{BCD}
∠
B
A
D
=
2∠
BC
D
and
A
B
=
A
D
AB = AD
A
B
=
A
D
. Let
P
P
P
be a point such that
A
B
C
P
ABCP
A
BCP
is a parallelogram. Prove that
C
P
=
D
P
CP = DP
CP
=
D
P
.
10.1
1
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Sharygin Geometry Olympiad 2022 Final Grade10 P1
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
and
B
1
B
2
B
3
B
4
B_1B_2B_3B_4
B
1
B
2
B
3
B
4
are two squares with their vertices arranged clockwise.The perpendicular bisector of segment
A
1
B
1
,
A
2
B
2
,
A
3
B
3
,
A
4
B
4
A_1B_1,A_2B_2,A_3B_3,A_4B_4
A
1
B
1
,
A
2
B
2
,
A
3
B
3
,
A
4
B
4
and the perpendicular bisector of segment
A
2
B
2
,
A
3
B
3
,
A
4
B
4
,
A
1
B
1
A_2B_2,A_3B_3,A_4B_4,A_1B_1
A
2
B
2
,
A
3
B
3
,
A
4
B
4
,
A
1
B
1
intersect at point
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
respectively.Show that:
P
R
⊥
Q
S
PR\perp QS
PR
⊥
QS
.
10.2
1
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Conditional geo with quadrilateral and external tamgents
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. The common external tangents to circles
(
A
B
C
)
(ABC)
(
A
BC
)
and
(
A
C
D
)
(ACD)
(
A
C
D
)
meet at point
E
E
E
, the common external tangents to circles
(
A
B
D
)
(ABD)
(
A
B
D
)
and
(
B
C
D
)
(BCD)
(
BC
D
)
meet at point
F
F
F
. Let
F
F
F
lie on
A
C
AC
A
C
, prove that
E
E
E
lies on
B
D
BD
B
D
.
10.4
1
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Common internal tangent passes through a midpoint of a diagonal
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
B
=
∠
D
\angle B= \angle D
∠
B
=
∠
D
. Prove that the midpoint of
B
D
BD
B
D
lies on the common internal tangent to the incircles of triangles
A
B
C
ABC
A
BC
and
A
C
D
ACD
A
C
D
.
9.8
1
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five points on each circle, five circles through each point
Several circles are drawn on the plane and all points of their intersection or touching are marked. Is it possible that each circle contains exactly five marked points and each point belongs to exactly five circles?
9.7
1
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KR is perpendicular to a median in ABC
Let
H
H
H
be the orthocenter of an acute-angled triangle
A
B
C
ABC
A
BC
. The circumcircle of triangle
A
H
C
AHC
A
H
C
meets segments
A
B
AB
A
B
and
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
. Lines
P
Q
PQ
PQ
and
A
C
AC
A
C
meet at point
R
R
R
. A point
K
K
K
lies on the line
P
H
PH
P
H
in such a way that
∠
K
A
C
=
9
0
∘
\angle KAC = 90^{\circ}
∠
K
A
C
=
9
0
∘
. Prove that
K
R
KR
K
R
is perpendicular to one of the medians of triangle
A
B
C
ABC
A
BC
.
9.6
1
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PQ is perpendicular to line passing through circumcenters
Lateral sidelines
A
B
AB
A
B
and
C
D
CD
C
D
of a trapezoid
A
B
C
D
ABCD
A
BC
D
(
A
D
>
B
C
AD >BC
A
D
>
BC
) meet at point
P
P
P
. Let
Q
Q
Q
be a point of segment
A
D
AD
A
D
such that
B
Q
=
C
Q
BQ = CQ
BQ
=
CQ
. Prove that the line passing through the circumcenters of triangles
A
Q
C
AQC
A
QC
and
B
Q
D
BQD
BQ
D
is perpendicular to
P
Q
PQ
PQ
.
9.5
1
Hide problems
Cute angle chasing problem
Chords
A
B
AB
A
B
and
C
D
CD
C
D
of a circle
ω
\omega
ω
meet at point
E
E
E
in such a way that
A
D
=
A
E
=
E
B
AD = AE = EB
A
D
=
A
E
=
EB
. Let
F
F
F
be a point of segment
C
E
CE
CE
such that
E
D
=
C
F
ED = CF
E
D
=
CF
. The bisector of angle
A
F
C
AFC
A
FC
meets an arc
D
A
C
DAC
D
A
C
at point
P
P
P
. Prove that
A
A
A
,
E
E
E
,
F
F
F
, and
P
P
P
are concyclic.
9.4
1
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Lines intersect on angle bisector
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
,
P
P
P
be the midpoint of the minor arc
A
B
AB
A
B
of its circumcircle, and
Q
Q
Q
be the midpoint of
A
C
AC
A
C
. A circumcircle of triangle
A
P
Q
APQ
A
PQ
centered at
O
O
O
meets
A
B
AB
A
B
for the second time at point
K
K
K
. Prove that lines
P
O
PO
PO
and
K
Q
KQ
K
Q
meet on the bisector of angle
A
B
C
ABC
A
BC
.
9.3
1
Hide problems
Spoiler alert: √(ac/2) inversion!
A medial line parallel to the side
A
C
AC
A
C
of triangle
A
B
C
ABC
A
BC
meets its circumcircle at points at
X
X
X
and
Y
Y
Y
. Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
and
D
D
D
be the midpoint of arc
A
C
AC
A
C
not containing
B
B
B
.A point
L
L
L
lie on segment
D
I
DI
D
I
in such a way that
D
L
=
B
I
/
2
DL= BI/2
D
L
=
B
I
/2
. Prove that
∠
I
X
L
=
∠
I
Y
L
\angle IXL = \angle IYL
∠
I
X
L
=
∠
I
Y
L
.
9.2
1
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Maximize (AP_1)(AP_2)
Let circles
s
1
s_1
s
1
and
s
2
s_2
s
2
meet at points
A
A
A
and
B
B
B
. Consider all lines passing through
A
A
A
and meeting the circles for the second time at points
P
1
P_1
P
1
and
P
2
P_2
P
2
respectively. Construct by a compass and a ruler a line such that
A
P
1
.
A
P
2
AP_1.AP_2
A
P
1
.
A
P
2
is maximal.
9.1
1
Hide problems
Excircles produce homothetic lines
Let
B
H
BH
B
H
be an altitude of right angled triangle
A
B
C
ABC
A
BC
(
∠
B
=
9
0
o
\angle B = 90^o
∠
B
=
9
0
o
). An excircle of triangle
A
B
H
ABH
A
B
H
opposite to
B
B
B
touches
A
B
AB
A
B
at point
A
1
A_1
A
1
; a point
C
1
C_1
C
1
is defined similarly. Prove that
A
C
/
/
A
1
C
1
AC // A_1C_1
A
C
//
A
1
C
1
.
24
1
Hide problems
Hail to the King - coplanar lines given and wanted, hexagonal pyramid, sphere
Let
O
A
B
C
D
E
F
OABCDEF
O
A
BC
D
EF
be a hexagonal pyramid with base
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
circumscribed around a sphere
ω
\omega
ω
. The plane passing through the touching points of
ω
\omega
ω
with faces
O
F
A
OFA
OF
A
,
O
A
B
OAB
O
A
B
and
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
meets
O
A
OA
O
A
at point
A
1
A_1
A
1
, points
B
1
B_1
B
1
,
C
1
C_1
C
1
,
D
1
D_1
D
1
,
E
1
E_1
E
1
and
F
1
F_1
F
1
are defined similarly. Let
ℓ
\ell
ℓ
,
m
m
m
and
n
n
n
be the lines
A
1
D
1
A_1D_1
A
1
D
1
,
B
1
E
1
B_1E_1
B
1
E
1
and
C
1
F
1
C_1F_1
C
1
F
1
respectively. It is known that
ℓ
\ell
ℓ
and
m
m
m
are coplanar, also
m
m
m
and
n
n
n
are coplanar. Prove that
ℓ
\ell
ℓ
and
n
n
n
are coplanar.
5
1
Hide problems
tangent of (DP_1A_1) // BC, cyclic ABCD
Let the diagonals of cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
meet at point
P
P
P
. The line passing through
P
P
P
and perpendicular to
P
D
PD
P
D
meets
A
D
AD
A
D
at point
D
1
D_1
D
1
, a point
A
1
A_1
A
1
is defined similarly. Prove that the tangent at
P
P
P
to the circumcircle of triangle
D
1
P
A
1
D_1PA_1
D
1
P
A
1
is parallel to
B
C
BC
BC
.
4
1
Hide problems
concurrency wanted, touchpoints with incircle, altitudes related
Let
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
,
C
C
1
CC_1
C
C
1
be the altitudes of acute angled triangle
A
B
C
ABC
A
BC
.
A
2
A_2
A
2
be the touching point of the incircle of triangle
A
B
1
C
1
AB_1C_1
A
B
1
C
1
with
B
1
C
1
B_1C_1
B
1
C
1
, points
B
2
B_2
B
2
,
C
2
C_2
C
2
be defined similarly. Prove that the lines
A
1
A
2
A_1A_2
A
1
A
2
,
B
1
B
2
B_1B_2
B
1
B
2
,
C
1
C
2
C_1C_2
C
1
C
2
concur.
3
1
Hide problems
FL = CL + LD wanted, altitude of right triangle ABC, equilaterals AED, CFD
Let
C
D
CD
C
D
be an altitude of right-angled triangle
A
B
C
ABC
A
BC
with
∠
C
=
90
\angle C = 90
∠
C
=
90
. Regular triangles
A
E
D
AED
A
E
D
and
C
F
D
CFD
CF
D
are such that
E
E
E
lies on the same side from
A
B
AB
A
B
as
C
C
C
, and
F
F
F
lies on the same side from
C
D
CD
C
D
as
B
B
B
. The line
E
F
EF
EF
meets
A
C
AC
A
C
at
L
L
L
. Prove that
F
L
=
C
L
+
L
D
FL = CL + LD
F
L
=
C
L
+
L
D
12
1
Hide problems
Questionable parallelogram
Let
K
K
K
,
L
L
L
,
M
M
M
,
N
N
N
be the midpoints of sides
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
,
A
B
AB
A
B
respectively of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. The common points of segments
A
K
AK
A
K
,
B
L
BL
B
L
,
C
M
CM
CM
,
D
N
DN
D
N
divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that
A
B
C
D
ABCD
A
BC
D
is a parallelogram?
14
1
Hide problems
12 altitudes madness
A triangle
A
B
C
ABC
A
BC
is given. Let
C
′
C'
C
′
and
C
a
′
C'_{a}
C
a
′
be the touching points of sideline
A
B
AB
A
B
with the incircle and with the excircle touching the side
B
C
BC
BC
. Points
C
b
′
C'_{b}
C
b
′
,
C
c
′
C'_{c}
C
c
′
,
A
′
A'
A
′
,
A
a
′
A'_{a}
A
a
′
,
A
b
′
A'_{b}
A
b
′
,
A
c
′
A'_{c}
A
c
′
,
B
′
B'
B
′
,
B
a
′
B'_{a}
B
a
′
,
B
b
′
B'_{b}
B
b
′
,
B
c
′
B'_{c}
B
c
′
are defined similarly. Consider the lengths of
12
12
12
altitudes of triangles
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
,
A
a
′
B
a
′
C
a
′
A'_{a}B'_{a}C'_{a}
A
a
′
B
a
′
C
a
′
,
A
b
′
B
b
′
C
b
′
A'_{b}B'_{b}C'_{b}
A
b
′
B
b
′
C
b
′
,
A
c
′
B
c
′
C
c
′
A'_{c}B'_{c}C'_{c}
A
c
′
B
c
′
C
c
′
. (a) (8-9) Find the maximal number of different lengths. (b) (10-11) Find all possible numbers of different lengths.
2
1
Hide problems
Circumcenter lies on bisector ...
Let
A
B
C
D
ABCD
A
BC
D
be a curcumscribed quadrilateral with incenter
I
I
I
, and let
O
1
,
O
2
O_{1}, O_{2}
O
1
,
O
2
be the circumcenters of triangles
A
I
D
AID
A
I
D
and
C
I
D
CID
C
I
D
. Prove that the circumcenter of triangle
O
1
I
O
2
O_{1}IO_{2}
O
1
I
O
2
lies on the bisector of angle
A
B
C
ABC
A
BC
1
1
Hide problems
Sharygin 2022 - P1
Let
O
O
O
and
H
H
H
be the circumcenter and the orthocenter respectively of triangle
A
B
C
ABC
A
BC
. Itis known that
B
H
BH
B
H
is the bisector of angle
A
B
O
ABO
A
BO
. The line passing through
O
O
O
and parallel to
A
B
AB
A
B
meets
A
C
AC
A
C
at
K
K
K
. Prove that
A
H
=
A
K
AH = AK
A
H
=
A
K
17
1
Hide problems
All lines PQ concur
Let a point
P
P
P
lie inside a triangle
A
B
C
ABC
A
BC
. The rays starting at
P
P
P
and crossing the sides
B
C
BC
BC
,
A
C
AC
A
C
,
A
B
AB
A
B
under the right angle meet the circumcircle of
A
B
C
ABC
A
BC
at
A
1
A_{1}
A
1
,
B
1
B_{1}
B
1
,
C
1
C_{1}
C
1
respectively. It is known that lines
A
A
1
AA_{1}
A
A
1
,
B
B
1
BB_{1}
B
B
1
,
C
C
1
CC_{1}
C
C
1
concur at point
Q
Q
Q
. Prove that all such lines
P
Q
PQ
PQ
concur.
20
1
Hide problems
Too many perpendicular and parallel lines
Let
O
O
O
,
I
I
I
be the circumcenter and the incenter of
△
A
B
C
\triangle ABC
△
A
BC
;
R
R
R
,
r
r
r
be the circumradius and the inradius;
D
D
D
be the touching point of the incircle with
B
C
BC
BC
; and
N
N
N
be an arbitrary point of segment
I
D
ID
I
D
. The perpendicular to
I
D
ID
I
D
at
N
N
N
meets the circumcircle of
A
B
C
ABC
A
BC
at points
X
X
X
and
Y
Y
Y
. Let
O
1
O_{1}
O
1
be the circumcircle of
△
X
I
Y
\triangle XIY
△
X
I
Y
. Find the product
O
O
1
⋅
I
N
OO_{1}\cdot IN
O
O
1
⋅
I
N
.
13
1
Hide problems
Cute Combo Geo
Eight points in a general position are given in the plane. The areas of all
56
56
56
triangles with vertices at these points are written in a row. Prove that it is possible to insert the symbols "
+
+
+
" and "
−
-
−
" between them in such a way that the obtained sum is equal to zero.
16
1
Hide problems
Feet of angle bisectors and two vertices are Concyclic!!
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral,
E
=
A
C
∩
B
D
E = AC \cap BD
E
=
A
C
∩
B
D
,
F
=
A
D
∩
B
C
F = AD \cap BC
F
=
A
D
∩
BC
. The bisectors of angles
A
F
B
AFB
A
FB
and
A
E
B
AEB
A
EB
meet
C
D
CD
C
D
at points
X
,
Y
X, Y
X
,
Y
. Prove that
A
,
B
,
X
,
Y
A, B, X, Y
A
,
B
,
X
,
Y
are concyclic.
22
1
Hide problems
Classic Geo Overload
Chords
A
1
A
2
,
A
3
A
4
,
A
5
A
6
A_1A_2, A_3A_4, A_5A_6
A
1
A
2
,
A
3
A
4
,
A
5
A
6
of a circle
Ω
\Omega
Ω
concur at point
O
O
O
. Let
B
i
B_i
B
i
be the second common point of
Ω
\Omega
Ω
and the circle with diameter
O
A
i
OA_i
O
A
i
. Prove that chords
B
1
B
2
,
B
3
B
4
,
B
5
B
6
B_1B_2, B_3B_4, B_5B_6
B
1
B
2
,
B
3
B
4
,
B
5
B
6
concur.
23
1
Hide problems
Quad sandwiched between confocal ellipses
An ellipse with focus
F
F
F
is given. Two perpendicular lines passing through
F
F
F
meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Prove that this quadrilateral is inscribed into a conic with focus
F
F
F
21
1
Hide problems
Bicentric Quadrilateral strikes again
The circumcenter
O
O
O
, the incenter
I
I
I
, and the midpoint
M
M
M
of a diagonal of a bicentral quadrilateral were marked. After this the quadrilateral was erased. Restore it.
19
1
Hide problems
Sharygin 2022 P19
Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
, and
K
K
K
be the common point of
B
C
BC
BC
with the external bisector of angle
A
A
A
. The line
K
I
KI
K
I
meets the external bisectors of angles
B
B
B
and
C
C
C
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
18
1
Hide problems
Reflection about diagonal in harmonic quad
The products of the opposite sidelengths of a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
are equal. Let
B
′
B'
B
′
be the reflection of
B
B
B
about
A
C
AC
A
C
. Prove that the circle passing through
A
,
B
′
,
D
A,B', D
A
,
B
′
,
D
touches
A
C
AC
A
C
15
1
Hide problems
Sharygin 2022 P15
A line
l
l
l
parallel to the side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
touches its incircle and meets its circumcircle at points
D
D
D
and
E
E
E
. Let
I
I
I
be the incenter of
A
B
C
ABC
A
BC
. Prove that
A
I
2
=
A
D
⋅
A
E
AI^2 = AD \cdot AE
A
I
2
=
A
D
⋅
A
E
.
11
1
Hide problems
Wait! T is the Dumpty point!
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
6
0
o
\angle A=60^o
∠
A
=
6
0
o
and
T
T
T
be a point such that
∠
A
T
B
=
∠
B
T
C
=
∠
A
T
C
\angle ATB=\angle BTC=\angle ATC
∠
A
TB
=
∠
BTC
=
∠
A
TC
. A circle passing through
B
,
C
B,C
B
,
C
and
T
T
T
meets
A
B
AB
A
B
and
A
C
AC
A
C
for the second time at points
K
K
K
and
L
L
L
.Prove that the distances from
K
K
K
and
L
L
L
to
A
T
AT
A
T
are equal.
10
1
Hide problems
Yum yum symmedia(m)n!
Let
ω
1
\omega_1
ω
1
be the circumcircle of triangle
A
B
C
ABC
A
BC
and
O
O
O
be its circumcenter. A circle
ω
2
\omega_2
ω
2
touches the sides
A
B
,
A
C
AB, AC
A
B
,
A
C
, and touches the arc
B
C
BC
BC
of
ω
1
\omega_1
ω
1
at point
K
K
K
. Let
I
I
I
be the incenter of
A
B
C
ABC
A
BC
. Prove that the line
O
I
OI
O
I
contains the symmedian of triangle
A
I
K
AIK
A
I
K
.
9
1
Hide problems
P,N,S are collinear
The sides
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
and
D
A
DA
D
A
of quadrilateral
A
B
C
D
ABCD
A
BC
D
touch a circle with center
I
I
I
at points
K
,
L
,
M
K, L, M
K
,
L
,
M
and
N
N
N
respectively. Let
P
P
P
be an arbitrary point of line
A
I
AI
A
I
. Let
P
K
PK
P
K
meet
B
I
BI
B
I
at point
Q
,
Q
L
Q, QL
Q
,
Q
L
meet
C
I
CI
C
I
at point
R
R
R
, and
R
M
RM
RM
meet
D
I
DI
D
I
at point
S
S
S
. Prove that
P
,
N
P,N
P
,
N
and
S
S
S
are collinear.
8
1
Hide problems
Locus of orthocentre of PQR is the diameter || to AC.
Points
P
,
Q
,
R
P,Q,R
P
,
Q
,
R
lie on the sides
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
of triangle
A
B
C
ABC
A
BC
in such a way that
A
P
=
P
R
,
C
Q
=
Q
R
AP=PR, CQ=QR
A
P
=
PR
,
CQ
=
QR
. Let
H
H
H
be the orthocenter of triangle
P
Q
R
PQR
PQR
, and
O
O
O
be the circumcenter of triangle
A
B
C
ABC
A
BC
. Prove that
O
H
∣
∣
A
C
OH||AC
O
H
∣∣
A
C
.
7
1
Hide problems
Triangle beside $^2$ construction
A square with center
F
F
F
was constructed on the side
A
C
AC
A
C
of triangle
A
B
C
ABC
A
BC
outside it. After this, everything was erased except
F
F
F
and the midpoints
N
,
K
N,K
N
,
K
of sides
B
C
,
A
B
BC,AB
BC
,
A
B
. Restore the triangle.
6
1
Hide problems
Geometric inequality
The incircle and the excircle of triangle
A
B
C
ABC
A
BC
touch the side
A
C
AC
A
C
at points
P
P
P
and
Q
Q
Q
respectively. The lines
B
P
BP
BP
and
B
Q
BQ
BQ
meet the circumcircle of triangle
A
B
C
ABC
A
BC
for the second time at points
P
′
P'
P
′
and
Q
′
Q'
Q
′
respectively. Prove that
P
P
′
>
Q
Q
′
PP' > QQ'
P
P
′
>
Q
Q
′