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Problems
Contests
National and Regional Contests
Russia Contests
Sharygin Geometry Olympiad
2023 Sharygin Geometry Olympiad
2023 Sharygin Geometry Olympiad
Part of
Sharygin Geometry Olympiad
Subcontests
(48)
10.8
1
Hide problems
prove that the radical center of omega_2, omega_3, omega_4 is in the ratio 2:1
A triangle
A
B
C
ABC
A
BC
is given. Let
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
,
ω
3
\omega_3
ω
3
,
ω
4
\omega_4
ω
4
be circles centered at points
X
X
X
,
Y
Y
Y
,
Z
Z
Z
,
T
T
T
respectively such that each of lines
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
cuts off on them four equal chords. Prove that the centroid of
A
B
C
ABC
A
BC
divides the segment joining
X
X
X
and the radical center of
ω
2
\omega_2
ω
2
,
ω
3
\omega_3
ω
3
,
ω
4
\omega_4
ω
4
in the ratio
2
:
1
2:1
2
:
1
from
X
X
X
.
10.7
1
Hide problems
prove triangles obtained by the transpositions of vertices are identical
There are
43
43
43
points in the space:
3
3
3
yellow and
40
40
40
red. Any four of them are not coplanar. May the number of triangles with red vertices hooked with the triangle with yellow vertices be equal to
2023
2023
2023
? Yellow triangle is hooked with the red one if the boundary of the red triangle meet the part of the plane bounded by the yellow triangle at the unique point. The triangles obtained by the transpositions of vertices are identical.
10.6
1
Hide problems
prove that the external tangents to AEB and AED meet on AEC
Let
E
E
E
be the projection of the vertex
C
C
C
of a rectangle
A
B
C
D
ABCD
A
BC
D
to the diagonal
B
D
BD
B
D
. Prove that the common external tangents to the circles
A
E
B
AEB
A
EB
and
A
E
D
AED
A
E
D
meet on the circle
A
E
C
AEC
A
EC
.
10.5
1
Hide problems
prove that the line PQ bisects AD
The incircle of a triangle
A
B
C
ABC
A
BC
touches
B
C
BC
BC
at point
D
D
D
. Let
M
M
M
be the midpoint of arc
B
A
C
^
\widehat{BAC}
B
A
C
of the circumcircle, and
P
P
P
,
Q
Q
Q
be the projections of
M
M
M
to the external bisectors of angles
B
B
B
and
C
C
C
respectively. Prove that the line
P
Q
PQ
PQ
bisects
A
D
AD
A
D
.
10.4
1
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find locus of points A2
Let
A
B
C
ABC
A
BC
be a Poncelet triangle,
A
1
A_1
A
1
is the reflection of
A
A
A
about the incenter
I
I
I
,
A
2
A_2
A
2
is isogonally conjugated to
A
1
A_1
A
1
with respect to
A
B
C
ABC
A
BC
. Find the locus of points
A
2
A_2
A
2
.
10.3
1
Hide problems
prove that FOG = 180 - 2BAC
Let
ω
\omega
ω
be the circumcircle of triangle
A
B
C
ABC
A
BC
,
O
O
O
be its center,
A
′
A'
A
′
be the point of
ω
\omega
ω
opposite to
A
A
A
, and
D
D
D
be a point on a minor arc
B
C
BC
BC
of
ω
\omega
ω
. A point
D
′
D'
D
′
is the reflection of
D
D
D
about
B
C
BC
BC
. The line
A
′
D
′
A'D'
A
′
D
′
meets for the second time at point
E
E
E
. The perpendicular bisector to
D
′
E
D'E
D
′
E
meets
A
B
AB
A
B
and
A
C
AC
A
C
at points
F
F
F
and
G
G
G
respectively. Prove that
∠
F
O
G
=
18
0
∘
−
2
∠
B
A
C
\angle FOG = 180^\circ - 2\angle BAC
∠
FOG
=
18
0
∘
−
2∠
B
A
C
.
10.2
1
Hide problems
euler line touches incircle, prove triangle is obtuse
The Euler line of a scalene triangle touches its incircle. Prove that this triangle is obtuse-angled.
10.1
1
Hide problems
prove DBC is equal to ACD and BCM
Let
M
M
M
be the midpoint of cathetus
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
with right angle
A
A
A
. Point
D
D
D
lies on the median
A
N
AN
A
N
of triangle
A
M
C
AMC
A
MC
in such a way that the angles
A
C
D
ACD
A
C
D
and
B
C
M
BCM
BCM
are equal. Prove that the angle
D
B
C
DBC
D
BC
is also equal to these angles.
8.8
1
Hide problems
locus of common points of lines BC and EF
Two circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
meeting at point
A
A
A
and a line
a
a
a
are given. Let
B
C
BC
BC
be an arbitrary chord of
ω
2
\omega_2
ω
2
parallel to
a
a
a
, and
E
E
E
,
F
F
F
be the second common points of
A
B
AB
A
B
and
A
C
AC
A
C
respectively with
ω
1
\omega_1
ω
1
. Find the locus of common points of lines
B
C
BC
BC
and
E
F
EF
EF
.
8.7
1
Hide problems
restore ABC if A, P, W are given
The bisector of angle
A
A
A
of triangle
A
B
C
ABC
A
BC
meet its circumcircle
ω
\omega
ω
at point
W
W
W
. The circle
s
s
s
with diameter
A
H
AH
A
H
(
H
H
H
is the orthocenter of
A
B
C
ABC
A
BC
) meets
ω
\omega
ω
for the second time at point
P
P
P
. Restore the triangle
A
B
C
ABC
A
BC
if the points
A
A
A
,
P
P
P
,
W
W
W
are given.
8.6
1
Hide problems
paved by congruent figures bounded by n arcs of circles
For which
n
n
n
the plane may be paved by congruent figures bounded by
n
n
n
arcs of circles?
8.5
1
Hide problems
find BD if MO = a and OC = b
The median
C
M
CM
CM
and the altitude
A
H
AH
A
H
of an acute-angled triangle
A
B
C
ABC
A
BC
meet at point
O
O
O
. A point
D
D
D
lies outside the triangle in such a way that
A
O
C
D
AOCD
A
OC
D
is a parallelogram. Find the length of
B
D
BD
B
D
, if
M
O
=
a
MO= a
MO
=
a
,
O
C
=
b
OC = b
OC
=
b
.
8.4
1
Hide problems
BM median, prove ME = MF
Let
A
B
C
ABC
A
BC
be an acute-angled triangle,
O
O
O
be its circumcenter,
B
M
BM
BM
be a median, and
B
H
BH
B
H
be an altitude. Circles
A
O
B
AOB
A
OB
and
B
H
C
BHC
B
H
C
meet for the second time at point
E
E
E
, and circles
A
H
B
AHB
A
H
B
and
B
O
C
BOC
BOC
meet at point
F
F
F
. Prove that
M
E
=
M
F
ME = MF
ME
=
MF
.
8.3
1
Hide problems
can unit square covered by parallelogram
The altitudes of a parallelogram are greater than
1
1
1
. Does this yield that the unit square may be covered by this parallelogram?
8.2
1
Hide problems
prove A1B1C1 and A2B2C2 are similar
The bisectors of angles
A
A
A
,
B
B
B
, and
C
C
C
of triangle
A
B
C
ABC
A
BC
meet for the second time its circumcircle at points
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
respectively. Let
A
2
A_2
A
2
,
B
2
B_2
B
2
,
C
2
C_2
C
2
be the midpoints of segments
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
,
C
C
1
CC_1
C
C
1
respectively. 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.
8.1
1
Hide problems
ABC isosceles obtuse, find ACD
Let
A
B
C
ABC
A
BC
be an isosceles obtuse-angled triangle, and
D
D
D
be a point on its base
A
B
AB
A
B
such that
A
D
AD
A
D
equals to the circumradius of triangle
B
C
D
BCD
BC
D
. Find the value of
∠
A
C
D
\angle ACD
∠
A
C
D
.
9.8
1
Hide problems
A=120, find value of PIQ
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
12
0
∘
\angle A = 120^\circ
∠
A
=
12
0
∘
,
I
I
I
be the incenter, and
M
M
M
be the midpoint of
B
C
BC
BC
. The line passing through
M
M
M
and parallel to
A
I
AI
A
I
meets the circle with diameter
B
C
BC
BC
at points
E
E
E
and
F
F
F
(
A
A
A
and
E
E
E
lie on the same semiplane with respect to
B
C
BC
BC
). The line passing through
E
E
E
and perpendicular to
F
I
FI
F
I
meets
A
B
AB
A
B
and
A
C
AC
A
C
at points
P
P
P
and
Q
Q
Q
respectively. Find the value of
∠
P
I
Q
\angle PIQ
∠
P
I
Q
.
9.7
1
Hide problems
prove line joining vertices of T_1 and H are perpendicular to sidelines of T_2
Let
H
H
H
be the orthocenter of triangle
T
\mathrm T
T
. The sidelines of triangle
T
1
\mathrm T_1
T
1
pass through the midpoints of
T
\mathrm T
T
and are perpendicular to the corresponding bisectors of
T
\mathrm T
T
. The vertices of triangle
T
2
\mathrm T_2
T
2
bisect the bisectors of
T
\mathrm T
T
. Prove that the lines joining
H
H
H
with the vertices of
T
1
\mathrm T_1
T
1
are perpendicular to the sidelines of
T
2
\mathrm T_2
T
2
.
9.6
1
Hide problems
prove AB'C' and PQN are tangent
Let
A
B
C
ABC
A
BC
be acute-angled triangle with circumcircle
Γ
\Gamma
Γ
. Points
H
H
H
and
M
M
M
are the orthocenter and the midpoint of
B
C
BC
BC
respectively. The line
H
M
HM
H
M
meets the circumcircle
ω
\omega
ω
of triangle
B
H
C
BHC
B
H
C
at point
N
≠
H
N\not= H
N
=
H
. Point
P
P
P
lies on the arc
B
C
BC
BC
of
ω
\omega
ω
not containing
H
H
H
in such a way that
∠
H
M
P
=
9
0
∘
\angle HMP = 90^\circ
∠
H
MP
=
9
0
∘
. The segment
P
M
PM
PM
meets
Γ
\Gamma
Γ
at point
Q
Q
Q
. Points
B
′
B'
B
′
and
C
′
C'
C
′
are the reflections of
A
A
A
about
B
B
B
and
C
C
C
respectively. Prove that the circumcircles of triangles
A
B
′
C
′
AB'C'
A
B
′
C
′
and
P
Q
N
PQN
PQN
are tangent.
9.5
1
Hide problems
prove tangent to (ABD) at B bisects EC
A point
D
D
D
lie on the lateral side
B
C
BC
BC
of an isosceles triangle
A
B
C
ABC
A
BC
. The ray
A
D
AD
A
D
meets the line passing through
B
B
B
and parallel to the base
A
C
AC
A
C
at point
E
E
E
. Prove that the tangent to the circumcircle of triangle
A
B
D
ABD
A
B
D
at
B
B
B
bisects
E
C
EC
EC
.
9.4
1
Hide problems
prove AIP and omega cut off equal chords on AD
The incircle
ω
\omega
ω
of a triangle
A
B
C
ABC
A
BC
centered at
I
I
I
touches
B
C
BC
BC
at point
D
D
D
. Let
P
P
P
be the projection of the orthocenter of
A
B
C
ABC
A
BC
to the median from
A
A
A
. Prove that the circle
A
I
P
AIP
A
I
P
and
ω
\omega
ω
cut off equal chords on
A
D
AD
A
D
.
9.3
1
Hide problems
prove all lines A'B' concur
Points
A
1
A_1
A
1
,
A
2
A_2
A
2
,
B
1
B_1
B
1
,
B
2
B_2
B
2
lie on the circumcircle of a triangle
A
B
C
ABC
A
BC
in such a way that
A
1
B
1
∥
A
B
A_1B_1 \parallel AB
A
1
B
1
∥
A
B
,
A
1
A
2
∥
B
C
A_1A_2 \parallel BC
A
1
A
2
∥
BC
,
B
1
B
2
∥
A
C
B_1B_2 \parallel AC
B
1
B
2
∥
A
C
. The line
A
A
2
AA_2
A
A
2
and
C
A
1
CA_1
C
A
1
meet at point
A
′
A'
A
′
, and the lines
B
B
2
BB_2
B
B
2
and
C
B
1
CB_1
C
B
1
meet at point
B
′
B'
B
′
. Prove that all lines
A
′
B
′
A'B'
A
′
B
′
concur.
9.2
1
Hide problems
no internal points in regular triangle inside regular hexagon
Can a regular triangle be placed inside a regular hexagon in such a way that all vertices of the triangle were seen from each vertex of the hexagon? (Point
A
A
A
is seen from
B
B
B
, if the segment
A
B
AB
A
B
dots not contain internal points of the triangle.)
9.1
1
Hide problems
4 of the 6 trisector points are concyclic
The ratio of the median
A
M
AM
A
M
of a triangle
A
B
C
ABC
A
BC
to the side
B
C
BC
BC
equals
3
:
2
\sqrt{3}:2
3
:
2
. The points on the sides of
A
B
C
ABC
A
BC
dividing these side into
3
3
3
equal parts are marked. Prove that some
4
4
4
of these
6
6
6
points are concyclic.
24
1
Hide problems
concyclicity of second intersections with sphere
A tetrahedron
A
B
C
D
ABCD
A
BC
D
is give. A line
ℓ
\ell
ℓ
meets the planes
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
at points
D
0
,
A
0
,
B
0
,
C
0
D_0,A_0,B_0,C_0
D
0
,
A
0
,
B
0
,
C
0
respectively. Let
P
P
P
be an arbitrary point not lying on
ℓ
\ell
ℓ
and the planes of the faces, and
A
1
,
B
1
,
C
1
,
D
1
A_1,B_1,C_1,D_1
A
1
,
B
1
,
C
1
,
D
1
be the second common points of lines
P
A
0
,
P
B
0
,
P
C
0
,
P
D
0
PA_0,PB_0,PC_0,PD_0
P
A
0
,
P
B
0
,
P
C
0
,
P
D
0
with the spheres
P
B
C
D
,
P
C
D
A
,
P
D
A
B
,
P
A
B
C
PBCD,PCDA,PDAB,PABC
PBC
D
,
PC
D
A
,
P
D
A
B
,
P
A
BC
respectively. Prove
P
,
A
1
,
B
1
,
C
1
,
D
1
P,A_1,B_1,C_1,D_1
P
,
A
1
,
B
1
,
C
1
,
D
1
lie on a circle.
23
1
Hide problems
common line of ellipses pass through orthocenter
An ellipse
Γ
1
\Gamma_1
Γ
1
with foci at the midpoints of sides
A
B
AB
A
B
and
A
C
AC
A
C
of a triangle
A
B
C
ABC
A
BC
passes through
A
A
A
, and an ellipse
Γ
2
\Gamma_2
Γ
2
with foci at the midpoints of
A
C
AC
A
C
and
B
C
BC
BC
passes through
C
C
C
. Prove that the common points of these ellipses and the orthocenter of triangle
A
B
C
ABC
A
BC
are collinear.
22
1
Hide problems
incircle, excircle tangent projective geo
Let
A
B
C
ABC
A
BC
be a scalene triangle,
M
M
M
be the midpoint of
B
C
,
P
BC,P
BC
,
P
be the common point of
A
M
AM
A
M
and the incircle of
A
B
C
ABC
A
BC
closest to
A
A
A
, and
Q
Q
Q
be the common point of the ray
A
M
AM
A
M
and the excircle farthest from
A
A
A
. The tangent to the incircle at
P
P
P
meets
B
C
BC
BC
at point
X
X
X
, and the tangent to the excircle at
Q
Q
Q
meets
B
C
BC
BC
at
Y
Y
Y
. Prove that
M
X
=
M
Y
MX=MY
MX
=
M
Y
.
21
1
Hide problems
radical axis get spammed using humpty points
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral;
M
a
c
M_{ac}
M
a
c
be the midpoint of
A
C
AC
A
C
;
H
d
,
H
b
H_d,H_b
H
d
,
H
b
be the orthocenters of
△
A
B
C
,
△
A
D
C
\triangle ABC,\triangle ADC
△
A
BC
,
△
A
D
C
respectively;
P
d
,
P
b
P_d,P_b
P
d
,
P
b
be the projections of
H
d
H_d
H
d
and
H
b
H_b
H
b
to
B
M
a
c
BM_{ac}
B
M
a
c
and
D
M
a
c
DM_{ac}
D
M
a
c
respectively. Define similarly
P
a
,
P
c
P_a,P_c
P
a
,
P
c
for the diagonal
B
D
BD
B
D
. Prove that
P
a
,
P
b
,
P
c
,
P
d
P_a,P_b,P_c,P_d
P
a
,
P
b
,
P
c
,
P
d
are concyclic.
20
1
Hide problems
line joining isogonals parallel to line through tangents
Let a point
D
D
D
lie on the median
A
M
AM
A
M
of a triangle
A
B
C
ABC
A
BC
. The tangents to the circumcircle of triangle
B
D
C
BDC
B
D
C
at points
B
B
B
and
C
C
C
meet at point
K
K
K
. Prove that
D
D
′
DD'
D
D
′
is parallel to
A
K
AK
A
K
, where
D
′
D'
D
′
is isogonally conjugated to
D
D
D
with respect to
A
B
C
ABC
A
BC
.
19
1
Hide problems
locus of coaxial circles intersection point
A cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
is given. An arbitrary circle passing through
C
C
C
and
D
D
D
meets
A
C
,
B
C
AC,BC
A
C
,
BC
at points
X
,
Y
X,Y
X
,
Y
respectively. Find the locus of common points of circles
C
A
Y
CAY
C
A
Y
and
C
B
X
CBX
CBX
.
18
1
Hide problems
restore bicentral quadrilateral from arc midpoints
Restore a bicentral quadrilateral
A
B
C
D
ABCD
A
BC
D
if the midpoints of the arcs
A
B
,
B
C
,
C
D
AB,BC,CD
A
B
,
BC
,
C
D
of its circumcircle are given.
17
1
Hide problems
tangent nagelians with common point
A common external tangent to circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
touches them at points
T
1
,
T
2
T_1, T_2
T
1
,
T
2
respectively. Let
A
A
A
be an arbitrary point on the extension of
T
1
T
2
T_1T_2
T
1
T
2
beyond
T
1
T_1
T
1
, and
B
B
B
be a point on the extension of
T
1
T
2
T_1T_2
T
1
T
2
beyond
T
2
T_2
T
2
such that
A
T
1
=
B
T
2
AT_1 = BT_2
A
T
1
=
B
T
2
. The tangents from
A
A
A
to
ω
1
\omega_1
ω
1
and from
B
B
B
to
ω
2
\omega_2
ω
2
distinct from
T
1
T
2
T_1T_2
T
1
T
2
meet at point
C
C
C
. Prove that all nagelians of triangles
A
B
C
ABC
A
BC
from
C
C
C
have a common point.
16
1
Hide problems
vertex reflection concyclic with line joining foots
Let
A
H
A
AH_A
A
H
A
and
B
H
B
BH_B
B
H
B
be the altitudes of a triangle
A
B
C
ABC
A
BC
. The line
H
A
H
B
H_AH_B
H
A
H
B
meets the circumcircle of
A
B
C
ABC
A
BC
at points
P
P
P
and
Q
Q
Q
. Let
A
′
A'
A
′
be the reflection of
A
A
A
about
B
C
BC
BC
, and
B
′
B'
B
′
be the reflection of
B
B
B
about
C
A
CA
C
A
. Prove that
A
′
,
B
′
,
P
,
Q
A',B', P,Q
A
′
,
B
′
,
P
,
Q
are concyclic.
15
1
Hide problems
median concurrency with equal lengths
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral. Points
X
X
X
and
Y
Y
Y
lie on the extensions beyond
D
D
D
of the sides
C
D
CD
C
D
and
A
D
AD
A
D
respectively in such a way that
D
X
=
A
B
DX = AB
D
X
=
A
B
and
D
Y
=
B
C
DY = BC
D
Y
=
BC
. Similarly points
Z
Z
Z
and
T
T
T
lie on the extensions beyond
B
B
B
of the sides
C
B
CB
CB
and
A
B
AB
A
B
respectively in such a way that
B
Z
=
A
D
BZ = AD
BZ
=
A
D
and
B
T
=
D
C
BT = DC
BT
=
D
C
. Let
M
1
M_1
M
1
be the midpoint of
X
Y
XY
X
Y
, and
M
2
M_2
M
2
be the midpoint of
Z
T
ZT
ZT
. Prove that the lines
D
M
1
,
B
M
2
DM_1, BM_2
D
M
1
,
B
M
2
and
A
C
AC
A
C
concur.
14
1
Hide problems
symmetric polygon has odd winding number
Suppose that a closed oriented polygonal line
L
\mathcal{L}
L
in the plane does not pass through a point
O
O
O
, and is symmetric with respect to
O
O
O
. Prove that the winding number of
L
\mathcal{L}
L
around
O
O
O
is odd.The winding number of
L
\mathcal{L}
L
around
O
O
O
is defined to be the following sum of the oriented angles divided by
2
π
2\pi
2
π
:
deg
O
L
:
=
∠
A
1
O
A
2
+
∠
A
2
O
A
3
+
⋯
+
∠
A
n
−
1
O
A
n
+
∠
A
n
O
A
1
2
π
.
\deg_O\mathcal{L} := \dfrac{\angle A_1OA_2+\angle A_2OA_3+\dots+\angle A_{n-1}OA_n+\angle A_nOA_1}{2\pi}.
de
g
O
L
:=
2
π
∠
A
1
O
A
2
+
∠
A
2
O
A
3
+
⋯
+
∠
A
n
−
1
O
A
n
+
∠
A
n
O
A
1
.
13
1
Hide problems
find the greater angle of trapezoid
The base
A
D
AD
A
D
of a trapezoid
A
B
C
D
ABCD
A
BC
D
is twice greater than the base
B
C
BC
BC
, and the angle
C
C
C
equals one and a half of the angle
A
A
A
. The diagonal
A
C
AC
A
C
divides angle
C
C
C
into two angles. Which of them is greater?
12
1
Hide problems
perpendicular bisectors featuring circles
Let
A
B
C
ABC
A
BC
be a triangle with obtuse angle
B
B
B
, and
P
,
Q
P, Q
P
,
Q
lie on
A
C
AC
A
C
in such a way that
A
P
=
P
B
,
B
Q
=
Q
C
AP = PB, BQ = QC
A
P
=
PB
,
BQ
=
QC
. The circle
B
P
Q
BPQ
BPQ
meets the sides
A
B
AB
A
B
and
B
C
BC
BC
at points
N
N
N
and
M
M
M
respectively.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
a
)
<
/
s
p
a
n
>
\qquad<span class='latex-bold'>(a)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
a
)
<
/
s
p
an
>
(grades 8-9) Prove that the distances from the common point
R
R
R
of
P
M
PM
PM
and
N
Q
NQ
NQ
to
A
A
A
and
C
C
C
are equal.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
b
)
<
/
s
p
a
n
>
\qquad<span class='latex-bold'>(b)</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
b
)
<
/
s
p
an
>
(grades 10-11) Let
B
R
BR
BR
meet
A
C
AC
A
C
at point
S
S
S
. Prove that
M
N
⊥
O
S
MN \perp OS
MN
⊥
OS
, where
O
O
O
is the circumcenter of
A
B
C
ABC
A
BC
.
11
1
Hide problems
prove orthocenter for antipodal point
Let
H
H
H
be the orthocenter of an acute-angled triangle
A
B
C
ABC
A
BC
;
E
E
E
,
F
F
F
be points on
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively, such that
A
E
H
F
AEHF
A
E
H
F
is a parallelogram;
X
,
Y
X, Y
X
,
Y
be the common points of the line
E
F
EF
EF
and the circumcircle
ω
\omega
ω
of triangle
A
B
C
ABC
A
BC
;
Z
Z
Z
be the point of
ω
\omega
ω
opposite to
A
A
A
. Prove that
H
H
H
is the orthocenter of triangle
X
Y
Z
XYZ
X
Y
Z
.
10
1
Hide problems
angle bisector circumcircle tangency
Altitudes
B
E
BE
BE
and
C
F
CF
CF
of an acute-angled triangle
A
B
C
ABC
A
BC
meet at point
H
H
H
. The perpendicular from
H
H
H
to
E
F
EF
EF
meets the line
ℓ
\ell
ℓ
passing through
A
A
A
and parallel to
B
C
BC
BC
at point
P
P
P
. The bisectors of two angles between
ℓ
\ell
ℓ
and
H
P
HP
H
P
meet
B
C
BC
BC
at points
S
S
S
and
T
T
T
. Prove that the circumcircles of triangles
A
B
C
ABC
A
BC
and
P
S
T
PST
PST
are tangent.
9
1
Hide problems
altitude foot and midpoint concyclicity
It is known that the reflection of the orthocenter of a triangle
A
B
C
ABC
A
BC
about its circumcenter lies on
B
C
BC
BC
. Let
A
1
A_1
A
1
be the foot of the altitude from
A
A
A
. Prove that
A
1
A_1
A
1
lies on the circle passing through the midpoints of the altitudes of
A
B
C
ABC
A
BC
.
8
1
Hide problems
measured length construction with straightedge
A triangle
A
B
C
ABC
A
BC
(
a
>
b
>
c
)
(a>b>c)
(
a
>
b
>
c
)
is given. Its incenter
I
I
I
and the touching points
K
,
N
K, N
K
,
N
of the incircle with
B
C
BC
BC
and
A
C
AC
A
C
respectively are marked. Construct a segment with length
a
−
c
a-c
a
−
c
using only a ruler and drawing at most three lines.
7
1
Hide problems
tangency of moving nine point circle
Let
A
A
A
be a fixed point of a circle
ω
\omega
ω
. Let
B
C
BC
BC
be an arbitrary chord of
ω
\omega
ω
passing through a fixed point
P
P
P
. Prove that the nine-points circles of triangles
A
B
C
ABC
A
BC
touch some fixed circle not depending on
B
C
BC
BC
.
6
1
Hide problems
isogonal mittenpunkt concurrency gets resurrected
Let
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
be the feet of altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
. The incircle of triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
touches
A
1
B
1
,
A
1
C
1
,
B
1
C
1
A_1B_1, A_1C_1, B_1C_1
A
1
B
1
,
A
1
C
1
,
B
1
C
1
at points
C
2
,
B
2
,
A
2
C_2, B_2, A_2
C
2
,
B
2
,
A
2
respectively. Prove that the lines
A
A
2
,
B
B
2
,
C
C
2
AA_2, BB_2, CC_2
A
A
2
,
B
B
2
,
C
C
2
concur at a point lying on the Euler line of triangle
A
B
C
ABC
A
BC
.
5
1
Hide problems
prove perpendicularity with equal lengths and midpoint
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. Points
E
E
E
and
F
F
F
lie on the sides
A
D
AD
A
D
and
C
D
CD
C
D
in such a way that
A
E
=
B
C
AE = BC
A
E
=
BC
and
A
B
=
C
F
AB = CF
A
B
=
CF
. Let
M
M
M
be the midpoint of
E
F
EF
EF
. Prove that
∠
A
M
C
=
9
0
∘
\angle AMC = 90^{\circ}
∠
A
MC
=
9
0
∘
.
4
1
Hide problems
line passes through incenter of isoceles triangle
Points
D
D
D
and
E
E
E
lie on the lateral sides
A
B
AB
A
B
and
B
C
BC
BC
respectively of an isosceles triangle
A
B
C
ABC
A
BC
in such a way that
∠
B
E
D
=
3
∠
B
D
E
\angle BED = 3\angle BDE
∠
BE
D
=
3∠
B
D
E
. Let
D
′
D'
D
′
be the reflection of
D
D
D
about
A
C
AC
A
C
. Prove that the line
D
′
E
D'E
D
′
E
passes through the incenter of
A
B
C
ABC
A
BC
.
3
1
Hide problems
maximum diameter less than medial line
A circle touches the lateral sides of a trapezoid
A
B
C
D
ABCD
A
BC
D
at points
B
B
B
and
C
C
C
, and its center lies on
A
D
AD
A
D
. Prove that the diameter of the circle is less than the medial line of the trapezoid.
2
1
Hide problems
perpendiculars from tangency spam inside rectangle
The diagonals of a rectangle
A
B
C
D
ABCD
A
BC
D
meet at point
E
E
E
. A circle centered at
E
E
E
lies inside the rectangle. Let
C
F
CF
CF
,
D
G
DG
D
G
,
A
H
AH
A
H
be the tangents to this circle from
C
C
C
,
D
D
D
,
A
A
A
; let
C
F
CF
CF
meet
D
G
DG
D
G
at point
I
I
I
,
E
I
EI
E
I
meet
A
D
AD
A
D
at point
J
J
J
, and
A
H
AH
A
H
meet
C
F
CF
CF
at point
L
L
L
. Prove that
L
J
LJ
L
J
is perpendicular to
A
D
AD
A
D
.
1
1
Hide problems
concyclicity squashed by projections
Let
L
L
L
be the midpoint of the minor arc
A
C
AC
A
C
of the circumcircle of an acute-angled triangle
A
B
C
ABC
A
BC
. A point
P
P
P
is the projection of
B
B
B
to the tangent at
L
L
L
to the circumcircle. Prove that
P
P
P
,
L
L
L
, and the midpoints of sides
A
B
AB
A
B
,
B
C
BC
BC
are concyclic.