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Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2024 Sharygin Geometry Olympiad
2024 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(48)
10.8
1
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Trangles with same circumradius
The common tangents to the circumcircle and an excircle of triangle
A
B
C
ABC
A
BC
meet
B
C
,
C
A
,
A
B
BC, CA,AB
BC
,
C
A
,
A
B
at points
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
and
A
2
,
B
2
,
C
2
A_2, B_2, C_2
A
2
,
B
2
,
C
2
respectively. The triangle
Δ
1
\Delta_1
Δ
1
is formed by the lines
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
, the triangle
Δ
2
\Delta_2
Δ
2
is formed by the lines
A
A
2
,
B
B
2
,
AA_2, BB_2,
A
A
2
,
B
B
2
,
and
C
C
2
CC_2
C
C
2
. Prove that the circumradii of these triangles are equal.
10.7
1
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Chevian circle of I, projections
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
6
0
∘
\angle A=60^\circ
∠
A
=
6
0
∘
;
A
D
AD
A
D
,
B
E
BE
BE
, and
C
F
CF
CF
be its bisectors;
P
,
Q
P, Q
P
,
Q
be the projections of
A
A
A
to
E
F
EF
EF
and
B
C
BC
BC
respectively; and
R
R
R
be the second common point of the circle
D
E
F
DEF
D
EF
with
A
D
AD
A
D
. Prove that
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
are collinear.
10.6
1
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Brocard point on median
A point
P
P
P
lies on one of medians of triangle
A
B
C
ABC
A
BC
in such a way that
∠
P
A
B
=
∠
P
B
C
=
∠
P
C
A
\angle PAB =\angle PBC =\angle PCA
∠
P
A
B
=
∠
PBC
=
∠
PC
A
. Prove that there exists a point
Q
Q
Q
on another median such that
∠
Q
B
A
=
∠
Q
C
B
=
∠
Q
A
C
\angle QBA=\angle QCB =\angle QAC
∠
QB
A
=
∠
QCB
=
∠
Q
A
C
.
10.5
1
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Incircle, squares and equal areas
The incircle of a right-angled triangle
A
B
C
ABC
A
BC
touches the hypothenuse
A
B
AB
A
B
at point
T
T
T
. The squares
A
T
M
P
ATMP
A
TMP
and
B
T
N
Q
BTNQ
BTNQ
lie outside the triangle. Prove that the areas of triangles
A
B
C
ABC
A
BC
and
T
P
Q
TPQ
TPQ
are equal.
10.4
1
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Midpoint of $YA^*$ lies on A-excircle
Let
I
I
I
be the incenter of a triangle
A
B
C
ABC
A
BC
. The lines passing through
A
A
A
and parallel to
B
I
,
C
I
BI, CI
B
I
,
C
I
meet the perpendicular bisector to
A
I
AI
A
I
at points
S
,
T
S, T
S
,
T
respectively. Let
Y
Y
Y
be the common point of
B
T
BT
BT
and
C
S
CS
CS
, and
A
∗
A^*
A
∗
be a point such that
B
I
C
A
∗
BICA^*
B
I
C
A
∗
is a parallelogram. Prove that the midpoint of segment
Y
A
∗
YA^*
Y
A
∗
lies on the excircle of the triangle touching the side
B
C
BC
BC
.
10.3
1
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BE + CF \geq 2EF
Let
B
E
BE
BE
and
C
F
CF
CF
be the bisectors of a triangle
A
B
C
ABC
A
BC
. Prove that
2
E
F
≤
B
F
+
C
E
2EF \leq BF + CE
2
EF
≤
BF
+
CE
.
10.2
1
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Combi 3D problem
For which greatest
n
n
n
there exists a convex polyhedron with
n
n
n
faces having the following property: for each face there exists a point outside the polyhedron such that the remaining
n
−
1
n - 1
n
−
1
faces are seen from this point?
10.1
1
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AF, DE meet on median
The diagonals of a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
meet at point
P
P
P
. The bisector of angle
A
B
D
ABD
A
B
D
meets
A
C
AC
A
C
at point
E
E
E
, and the bisector of angle
A
C
D
ACD
A
C
D
meets
B
D
BD
B
D
at point
F
F
F
. Prove that the lines
A
F
AF
A
F
and
D
E
DE
D
E
meet on the median of triangle
A
P
D
APD
A
P
D
.
9.8
1
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A, P, Q, E are concylcic; isogonal conjugate
Let points
P
P
P
and
Q
Q
Q
be isogonally conjugated with respect to a triangle
A
B
C
ABC
A
BC
. The line
P
Q
PQ
PQ
meets the circumcircle of
A
B
C
ABC
A
BC
at point
X
X
X
. The reflection of
B
C
BC
BC
about
P
Q
PQ
PQ
meets
A
X
AX
A
X
at point
E
E
E
. Prove that
A
,
P
,
Q
,
E
A, P, Q, E
A
,
P
,
Q
,
E
are concyclic.
9.7
1
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Locus of XZ \cap ZT
Let
P
P
P
and
Q
Q
Q
be arbitrary points on the side
B
C
BC
BC
of triangle ABC such that
B
P
=
C
Q
BP = CQ
BP
=
CQ
. The common points of segments
A
P
AP
A
P
and
A
Q
AQ
A
Q
with the incircle form a quadrilateral
X
Y
Z
T
XYZT
X
Y
ZT
. Find the locus of common points of diagonals of such quadrilaterals.
9.6
1
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If 1 point on incircle, then all 3
The incircle of a triangle
A
B
C
ABC
A
BC
centered at
I
I
I
touches the sides
B
C
,
C
A
BC, CA
BC
,
C
A
, and
A
B
AB
A
B
at points
A
1
,
B
1
,
A_1, B_1,
A
1
,
B
1
,
and
C
1
C_1
C
1
respectively. The excircle centered at
J
J
J
touches the side
A
C
AC
A
C
at point
B
2
B_2
B
2
and touches the extensions of
A
B
,
B
C
AB, BC
A
B
,
BC
at points
C
2
,
A
2
C_2, A_2
C
2
,
A
2
respectively. Let the lines
I
B
2
IB_2
I
B
2
and
J
B
1
JB_1
J
B
1
meet at point
X
X
X
, the lines
I
C
2
IC_2
I
C
2
and
J
C
1
JC_1
J
C
1
meet at point
Y
Y
Y
, the lines
I
A
2
IA_2
I
A
2
and
J
A
1
JA_1
J
A
1
meet at point
Z
Z
Z
. Prove that if one of points
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
lies on the incircle then two remaining points also lie on it.
9.5
1
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Brocard point in isosceles triangle
Let
A
B
C
ABC
A
BC
be an isosceles triangle
(
A
C
=
B
C
)
(AC = BC)
(
A
C
=
BC
)
,
O
O
O
be its circumcenter,
H
H
H
be the orthocenter, and
P
P
P
be a point inside the triangle such that
∠
A
P
H
=
∠
B
P
O
=
π
/
2
\angle APH = \angle BPO = \pi /2
∠
A
P
H
=
∠
BPO
=
π
/2
. Prove that
∠
P
A
C
=
∠
P
B
A
=
∠
P
C
B
\angle PAC = \angle PBA = \angle PCB
∠
P
A
C
=
∠
PB
A
=
∠
PCB
.
9.4
1
Hide problems
Combi geometry in Sharygin's olympiad 2024
For which
n
>
0
n > 0
n
>
0
it is possible to mark several different points and several different circles on the plane in such a way that: — exactly
n
n
n
marked circles pass through each marked point; — exactly
n
n
n
marked points lie on each marked circle; — the center of each marked circle is marked?
9.3
1
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2 pairs of isogonal conjugated points and coaxial circles
Let
(
P
,
P
′
)
(P, P')
(
P
,
P
′
)
and
(
Q
,
Q
′
)
(Q, Q')
(
Q
,
Q
′
)
be two pairs of points isogonally conjugated with respect to a triangle
A
B
C
ABC
A
BC
, and
R
R
R
be the common point of lines
P
Q
PQ
PQ
and
P
′
Q
′
P'Q'
P
′
Q
′
. Prove that the pedal circles of points
P
P
P
,
Q
Q
Q
, and
R
R
R
are coaxial.
9.2
1
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Triangle with sides AB+CD,... is acute
Points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
on the plane do not form a rectangle. Let the sidelengths of triangle
T
T
T
equal
A
B
+
C
D
AB+CD
A
B
+
C
D
,
A
C
+
B
D
AC+BD
A
C
+
B
D
,
A
D
+
B
C
AD+BC
A
D
+
BC
. Prove that the triangle
T
T
T
is acute-angled.
9.1
1
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Orthocenter, midpoints, incircle
Let
H
H
H
be the orthocenter of an acute-angled triangle
A
B
C
ABC
A
BC
;
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
be the touching points of the incircle with
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
respectively;
E
A
,
E
B
,
E
C
E_A, E_B, E_C
E
A
,
E
B
,
E
C
be the midpoints of
A
H
,
B
H
,
C
H
AH, BH, CH
A
H
,
B
H
,
C
H
respectively. The circle centered at
E
A
E_A
E
A
and passing through
A
A
A
meets for the second time the bisector of angle
A
A
A
at
A
2
A_2
A
2
; points
B
2
,
C
2
B_2, C_2
B
2
,
C
2
are defined similarly. 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 similar.
8.8
1
Hide problems
suspicious clever combigeo
Two polygons are cut from the cardboard. Is it possible that for any disposition of these polygons on the plane they have either common inner points or only a finite number of common points on the boundary?
8.7
1
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NO WAY - line intersects quad
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is given. A line
l
∥
A
C
l \parallel AC
l
∥
A
C
meets the lines
A
D
AD
A
D
,
B
C
BC
BC
,
A
B
AB
A
B
,
C
D
CD
C
D
at points
X
X
X
,
Y
Y
Y
,
Z
Z
Z
,
T
T
T
respectively. The circumcircles of triangles
X
Y
B
XYB
X
Y
B
and
Z
T
B
ZTB
ZTB
meet for the second time at point
R
R
R
. Prove that
R
R
R
lies on
B
D
BD
B
D
.
8.6
1
Hide problems
locus of projections from the circumcenter - returns
A circle
ω
\omega
ω
touched lines
a
a
a
and
b
b
b
at points
A
A
A
and
B
B
B
respectively. An arbitrary tangent to the circle meets
a
a
a
and
b
b
b
at
X
X
X
and
Y
Y
Y
respectively. Points
X
′
X'
X
′
and
Y
′
Y'
Y
′
are the reflections of
X
X
X
and
Y
Y
Y
about
A
A
A
and
B
B
B
respectively. Find the locus of projections of the center of the circle to the lines
X
′
Y
′
X'Y'
X
′
Y
′
.
8.5
1
Hide problems
rectangle with vertices on sides of triangle
The vertices
M
M
M
,
N
N
N
,
K
K
K
of rectangle
K
L
M
N
KLMN
K
L
MN
lie on the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
A
CA
C
A
respectively of a regular triangle
A
B
C
ABC
A
BC
in such a way that
A
M
=
2
AM = 2
A
M
=
2
,
K
C
=
1
KC = 1
K
C
=
1
. The vertex
L
L
L
lies outside the triangle. Find the value of
∠
K
M
N
\angle KMN
∠
K
MN
.
8.4
1
Hide problems
length construction using folds
A square with side
1
1
1
is cut from the paper. Construct a segment with length
1
/
2024
1/2024
1/2024
using at most
20
20
20
folds. No instruments are available. It is allowed only to fold the paper and to mark the common points of folding lines.
8.3
1
Hide problems
a concyclicity with the foot of altitude
Let
A
D
AD
A
D
be the altitude of an acute-angled triangle
A
B
C
ABC
A
BC
and
A
′
A'
A
′
be the point on its circumcircle opposite to
A
A
A
. A point
P
P
P
lies on the segment
A
D
AD
A
D
, and points
X
X
X
,
Y
Y
Y
lie on the segments
A
B
AB
A
B
,
A
C
AC
A
C
respectively in such a way that
∠
C
B
P
=
∠
A
D
Y
\angle CBP = \angle ADY
∠
CBP
=
∠
A
D
Y
,
∠
B
C
P
=
∠
A
D
X
\angle BCP = \angle ADX
∠
BCP
=
∠
A
D
X
. Let
P
A
′
PA'
P
A
′
meet
B
C
BC
BC
at point
T
T
T
. Prove that
D
D
D
,
X
X
X
,
Y
Y
Y
,
T
T
T
are concyclic.
8.2
1
Hide problems
orthocentral bisection
Let
C
M
CM
CM
be the median of an acute-angled triangle
A
B
C
ABC
A
BC
, and
P
P
P
be the projection of the orthocenter
H
H
H
to the bisector of
∠
C
\angle C
∠
C
. Prove that
M
P
MP
MP
bisects the segment
C
H
CH
C
H
.
8.1
1
Hide problems
all lines are parallel
A circle
ω
\omega
ω
centered at
O
O
O
and a point
P
P
P
inside it are given. Let
X
X
X
be an arbitrary point of
ω
\omega
ω
, the line
X
P
XP
XP
and the circle
X
O
P
XOP
XOP
meet
ω
\omega
ω
for a second time at points
X
1
X_1
X
1
,
X
2
X_2
X
2
respectively. Prove that all lines
X
1
X
2
X_1X_2
X
1
X
2
are parallel.
24
1
Hide problems
The 3D algebra problem of Shary this year
Let
S
A
B
C
SABC
S
A
BC
be a pyramid with right angles at the vertex
S
S
S
. Points
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
lie on the edges
S
A
,
S
B
,
S
C
SA, SB, SC
S
A
,
SB
,
SC
respectively in such a way that the triangles
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
are similar. Does this yield that the planes
A
B
C
ABC
A
BC
and
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
are parallel?
23
1
Hide problems
One trick problem?!
A point
P
P
P
moves along a circle
Ω
\Omega
Ω
. Let
A
A
A
and
B
B
B
be two fixed points of
Ω
\Omega
Ω
, and
C
C
C
be an arbitrary point inside
Ω
\Omega
Ω
. The common external tangents to the circumcircles of triangles
A
P
C
APC
A
PC
and
B
C
P
BCP
BCP
meet at point
Q
Q
Q
. Prove that all points
Q
Q
Q
lie on two fixed lines.
22
1
Hide problems
Locus of touch points between the ellipse and the line
A segment
A
B
AB
A
B
is given. Let
C
C
C
be an arbitrary point of the perpendicular bisector to
A
B
AB
A
B
;
O
O
O
be the point on the circumcircle of
A
B
C
ABC
A
BC
opposite to
C
C
C
; and an ellipse centred at
O
O
O
touch
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
. Find the locus of touching points of the ellipse with the line
B
C
BC
BC
.
21
1
Hide problems
I, Blaise Pascal, have a dream
A chord
P
Q
PQ
PQ
of the circumcircle of a triangle
A
B
C
ABC
A
BC
meets the sides
B
C
,
A
C
BC, AC
BC
,
A
C
at points
A
′
,
B
′
A', B'
A
′
,
B
′
respectively. The tangents to the circumcircle at
A
A
A
and
B
B
B
meet at a point
X
X
X
, and the tangents at points
P
P
P
and
Q
Q
Q
meet at point
Y
Y
Y
. The line
X
Y
XY
X
Y
meets
A
B
AB
A
B
at a point
C
′
C'
C
′
. Prove that the lines
A
A
′
,
B
B
′
AA', BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
concur.
20
1
Hide problems
How on earth do you write problems like these?
Lines
a
1
,
b
1
,
c
1
a_1, b_1, c_1
a
1
,
b
1
,
c
1
pass through the vertices
A
,
B
,
C
A, B, C
A
,
B
,
C
respectively of a triange
A
B
C
ABC
A
BC
;
a
2
,
b
2
,
c
2
a_2, b_2, c_2
a
2
,
b
2
,
c
2
are the reflections of
a
1
,
b
1
,
c
1
a_1, b_1, c_1
a
1
,
b
1
,
c
1
about the corresponding bisectors of
A
B
C
ABC
A
BC
;
A
1
=
b
1
∩
c
1
,
B
1
=
a
1
∩
c
1
,
C
1
=
a
1
∩
b
1
A_1 = b_1 \cap c_1, B_1 = a_1 \cap c_1, C_1 = a_1 \cap b_1
A
1
=
b
1
∩
c
1
,
B
1
=
a
1
∩
c
1
,
C
1
=
a
1
∩
b
1
, and
A
2
,
B
2
,
C
2
A_2, B_2, C_2
A
2
,
B
2
,
C
2
are defined similarly. 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
have the same ratios of the area and circumradius (i.e.
S
1
R
1
=
S
2
R
2
\frac{S_1}{R_1} = \frac{S_2}{R_2}
R
1
S
1
=
R
2
S
2
, where
S
i
=
S
(
△
A
i
B
i
C
i
)
S_i = S(\triangle A_iB_iC_i)
S
i
=
S
(
△
A
i
B
i
C
i
)
,
R
i
=
R
(
△
A
i
B
i
C
i
)
R_i = R(\triangle A_iB_iC_i)
R
i
=
R
(
△
A
i
B
i
C
i
)
)
19
1
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An incentre is all it takes to replace the compass
A triangle
A
B
C
ABC
A
BC
, its circumcircle, and its incenter
I
I
I
are drawn on the plane. Construct the circumcenter of
A
B
C
ABC
A
BC
using only a ruler.
18
1
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An interesting triangle center (?) or not
Let
A
A
1
,
B
B
1
,
C
C
1
AA_1, BB_1, CC_1
A
A
1
,
B
B
1
,
C
C
1
be the altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
;
I
a
I_a
I
a
be its excenter corresponding to
A
A
A
;
I
a
′
I_a'
I
a
′
be the reflection of
I
a
I_a
I
a
about the line
A
A
1
AA_1
A
A
1
. Points
I
b
′
,
I
c
′
I_b', I_c'
I
b
′
,
I
c
′
are defined similarily. Prove that lines
A
1
I
a
′
,
B
1
I
b
′
,
C
1
I
c
′
A_1I_a', B_1I_b', C_1I_c'
A
1
I
a
′
,
B
1
I
b
′
,
C
1
I
c
′
concur.
17
1
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Beautiful problem with the incircle
Let
A
B
C
ABC
A
BC
be a non-isosceles triangle,
ω
\omega
ω
be its incircle. Let
D
,
E
,
D, E,
D
,
E
,
and
F
F
F
be the points at which the incircle of
A
B
C
ABC
A
BC
touches the sides
B
C
,
C
A
,
BC, CA,
BC
,
C
A
,
and
A
B
AB
A
B
respectively. Let
M
M
M
be the point on ray
E
F
EF
EF
such that
E
M
=
A
B
EM = AB
EM
=
A
B
. Let
N
N
N
be the point on ray
F
E
FE
FE
such that
F
N
=
A
C
FN = AC
FN
=
A
C
. Let the circumcircles of
△
B
F
M
\triangle BFM
△
BFM
and
△
C
E
N
\triangle CEN
△
CEN
intersect
ω
\omega
ω
again at
S
S
S
and
T
T
T
respectively. Prove that
B
S
,
C
T
,
BS, CT,
BS
,
CT
,
and
A
D
AD
A
D
concur.
16
1
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Nonstandard incentre config
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 bisectors of a triangle
A
B
C
ABC
A
BC
. The segments
B
B
1
BB_1
B
B
1
and
A
1
C
1
A_1C_1
A
1
C
1
meet at point
D
D
D
. Let
E
E
E
be the projection of
D
D
D
to
A
C
AC
A
C
. Points
P
P
P
and
Q
Q
Q
on sides
A
B
AB
A
B
and
B
C
BC
BC
respectively are such that
E
P
=
P
D
,
E
Q
=
Q
D
EP = PD, EQ = QD
EP
=
P
D
,
EQ
=
Q
D
. Prove that
∠
P
D
B
1
=
∠
E
D
Q
\angle PDB_1 = \angle EDQ
∠
P
D
B
1
=
∠
E
D
Q
.
15
1
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Algebra strikes again in Shary
The difference of two angles of a triangle is greater than
9
0
∘
90^{\circ}
9
0
∘
. Prove that the ratio of its circumradius and inradius is greater than
4
4
4
.
14
1
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Feuerbach point in a right angled triangle.
The incircle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
, right angled at
C
C
C
, touches the circumcircle of its medial triangle at point
F
F
F
. Let
O
E
OE
OE
be the tangent to
ω
\omega
ω
from the midpoint
O
O
O
of the hypotenuse
A
B
AB
A
B
, distinct from
A
B
AB
A
B
. Prove that
C
E
=
C
F
CE = CF
CE
=
CF
.
13
1
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Is this hard, or am I just dumb?
Can an arbitrary polygon be cut into isosceles trapezoids?
12
1
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What's the config here??
The bisectors
A
A
1
,
C
C
1
AA_1, CC_1
A
A
1
,
C
C
1
of a triangle
A
B
C
ABC
A
BC
with
∠
B
=
6
0
∘
\angle B = 60^{\circ}
∠
B
=
6
0
∘
meet at point
I
I
I
. The circumcircles of triangles
A
B
C
,
A
1
I
C
1
ABC, A_1IC_1
A
BC
,
A
1
I
C
1
meet at point
P
P
P
. Prove that the line
P
I
PI
P
I
bisects the side
A
C
AC
A
C
.
11
1
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What are points P, Q doing here?
Let
M
,
N
M, N
M
,
N
be the midpoints of sides
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively of a triangle
A
B
C
ABC
A
BC
. The perpendicular bisector to the bisectrix
A
L
AL
A
L
meets the bisectrixes of angles
B
B
B
and
C
C
C
at points
P
P
P
and
Q
Q
Q
respectively. Prove that the common point of lines
P
M
PM
PM
and
Q
N
QN
QN
lies on the tangent to the circumcircle of
A
B
C
ABC
A
BC
at
A
A
A
.
10
1
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Lines AP, AK must be isogonal
Let
ω
\omega
ω
be the circumcircle of triangle
A
B
C
ABC
A
BC
. A point
T
T
T
on the line
B
C
BC
BC
is such that
A
T
AT
A
T
touches
ω
\omega
ω
. The bisector of angle
B
A
C
BAC
B
A
C
meets
B
C
BC
BC
and
ω
\omega
ω
at points
L
L
L
and
A
0
A_0
A
0
respectively. The line
T
A
0
TA_0
T
A
0
meets
ω
\omega
ω
at point
P
P
P
. The point
K
K
K
lies on the segment
B
C
BC
BC
in such a way that
B
L
=
C
K
BL = CK
B
L
=
C
K
. Prove that
∠
B
A
P
=
∠
C
A
K
\angle BAP = \angle CAK
∠
B
A
P
=
∠
C
A
K
.
9
1
Hide problems
The incircle of a trapezoid...
Let
A
B
C
D
ABCD
A
BC
D
(
A
D
∥
B
C
AD \parallel BC
A
D
∥
BC
) be a trapezoid circumscribed around a circle
ω
\omega
ω
, which touches the sides
A
B
,
B
C
,
C
D
,
AB, BC, CD,
A
B
,
BC
,
C
D
,
and
A
D
AD
A
D
at points
P
,
Q
,
R
,
S
P, Q, R, S
P
,
Q
,
R
,
S
respectively. The line passing through
P
P
P
and parallel to the bases of the trapezoid meets
Q
R
QR
QR
at point
X
X
X
. Prove that
A
B
,
Q
S
AB, QS
A
B
,
QS
and
D
X
DX
D
X
concur.
8
1
Hide problems
A pair of equal sides, and a pair of equal angles!
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral
∠
B
=
∠
D
\angle B = \angle D
∠
B
=
∠
D
and
A
D
=
C
D
AD = CD
A
D
=
C
D
. The incircle of triangle
A
B
C
ABC
A
BC
touches the sides
B
C
BC
BC
and
A
B
AB
A
B
at points
E
E
E
and
F
F
F
respectively. Prove that the midpoints of segments
A
C
,
B
D
,
A
E
,
AC, BD, AE,
A
C
,
B
D
,
A
E
,
and
C
F
CF
CF
are concyclic.
6
1
Hide problems
What's that locus?
A circle
ω
\omega
ω
and two points
A
,
B
A, B
A
,
B
of this circle are given. Let
C
C
C
be an arbitrary point on one of arcs
A
B
AB
A
B
of
ω
\omega
ω
;
C
L
CL
C
L
be the bisector of triangle
A
B
C
ABC
A
BC
; the circle
B
C
L
BCL
BC
L
meet
A
C
AC
A
C
at point
E
E
E
; and
C
L
CL
C
L
meet
B
E
BE
BE
at point
F
F
F
. Find the locus of circumcenters of triangles
A
F
C
AFC
A
FC
.
5
1
Hide problems
Circles obtained after reflections concur.
Points
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
are the reflections of vertices
A
,
B
,
C
A, B, C
A
,
B
,
C
about the opposite sidelines of triangle
A
B
C
ABC
A
BC
. Prove that the circles
A
B
′
C
′
,
A
′
B
C
′
,
AB'C', A'BC',
A
B
′
C
′
,
A
′
B
C
′
,
and
A
′
B
′
C
A'B'C
A
′
B
′
C
have a common point.
2
1
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Is this combinatorics?
Three different collinear points are given. What is the number of isosceles triangles such that these points are their circumcenter, incenter and excenter (in some order)?
4
1
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Incircle Angle chase geo
The incircle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
touches
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at points
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
respectively,
P
P
P
is an arbitrary point on
ω
\omega
ω
. The line
A
P
AP
A
P
meets the circumcircle of triangle
A
B
1
C
1
AB_1C_1
A
B
1
C
1
for the second time at point
A
2
A_2
A
2
. Points
B
2
B_2
B
2
and
C
2
C_2
C
2
are defined similarly. Prove that the circumcircle of triangle
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
touches
ω
\omega
ω
.
3
1
Hide problems
Sharygin 2024 Correspondence P3
Let
A
B
C
ABC
A
BC
be an acute-angled triangle, and
M
M
M
be the midpoint of the minor arc
B
C
BC
BC
of its circumcircle. A circle
ω
\omega
ω
touches the side
A
B
,
A
C
AB, AC
A
B
,
A
C
at points
P
,
Q
P, Q
P
,
Q
respectively and passes through
M
M
M
. Prove that
B
P
+
C
Q
=
P
Q
BP + CQ = PQ
BP
+
CQ
=
PQ
.
1
1
Hide problems
Easy Geo
Bisectors
A
I
AI
A
I
and
C
I
CI
C
I
meet the circumcircle of triangle
A
B
C
ABC
A
BC
at points
A
1
,
C
1
A_1, C_1
A
1
,
C
1
respectively. The circumcircle of triangle
A
I
C
1
AIC_1
A
I
C
1
meets
A
B
AB
A
B
at point
C
0
C_0
C
0
; point
A
0
A_0
A
0
is defined similarly. Prove that
A
0
,
A
1
,
C
0
,
C
1
A_0, A_1, C_0, C_1
A
0
,
A
1
,
C
0
,
C
1
are collinear.
7
1
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Restoring Bicentral Quadrilateral
Restore a bicentral quadrilateral if two opposite vertices and the incenter are given.