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International Contests
Iranian Geometry Olympiad
2019 Iranian Geometry Olympiad
2019 Iranian Geometry Olympiad
Part of
Iranian Geometry Olympiad
Subcontests
(5)
1
3
Hide problems
2019 IGO Elementary P1
There is a table in the shape of a
8
×
5
8\times 5
8
×
5
rectangle with four holes on its corners. After shooting a ball from points
A
,
B
A, B
A
,
B
and
C
C
C
on the shown paths, will the ball fall into any of the holes after 6 reflections? (The ball reflects with the same angle after contacting the table edges.)http://s5.picofile.com/file/8372960750/E01.pngProposed by Hirad Alipanah
2019 IGO Intermediate P1
Two circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively intersect each other at points
A
A
A
and
B
B
B
, and point
O
1
O_1
O
1
lies on
ω
2
\omega_2
ω
2
. Let
P
P
P
be an arbitrary point lying on
ω
1
\omega_1
ω
1
. Lines
B
P
,
A
P
BP, AP
BP
,
A
P
and
O
1
O
2
O_1O_2
O
1
O
2
cut
ω
2
\omega_2
ω
2
for the second time at points
X
X
X
,
Y
Y
Y
and
C
C
C
, respectively. Prove that quadrilateral
X
P
Y
C
XPYC
XP
Y
C
is a parallelogram.Proposed by Iman Maghsoudi
2019 IGO Advanced P1
Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
intersect each other at points
A
A
A
and
B
B
B
. Point
C
C
C
lies on the tangent line from
A
A
A
to
ω
1
\omega_1
ω
1
such that
∠
A
B
C
=
9
0
∘
\angle ABC = 90^\circ
∠
A
BC
=
9
0
∘
. Arbitrary line
ℓ
\ell
ℓ
passes through
C
C
C
and cuts
ω
2
\omega_2
ω
2
at points
P
P
P
and
Q
Q
Q
. Lines
A
P
AP
A
P
and
A
Q
AQ
A
Q
cut
ω
1
\omega_1
ω
1
for the second time at points
X
X
X
and
Z
Z
Z
respectively. Let
Y
Y
Y
be the foot of altitude from
A
A
A
to
ℓ
\ell
ℓ
. Prove that points
X
,
Y
X, Y
X
,
Y
and
Z
Z
Z
are collinear.Proposed by Iman Maghsoudi
5
3
Hide problems
2019 IGO Elementary P5
For a convex polygon (i.e. all angles less than
18
0
∘
180^\circ
18
0
∘
) call a diagonal bisector if its bisects both area and perimeter of the polygon. What is the maximum number of bisector diagonals for a convex pentagon?Proposed by Morteza Saghafian
2019 IGO Intermediate P5
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
6
0
∘
\angle A = 60^\circ
∠
A
=
6
0
∘
. Points
E
E
E
and
F
F
F
are the foot of angle bisectors of vertices
B
B
B
and
C
C
C
respectively. Points
P
P
P
and
Q
Q
Q
are considered such that quadrilaterals
B
F
P
E
BFPE
BFPE
and
C
E
Q
F
CEQF
CEQF
are parallelograms. Prove that
∠
P
A
Q
>
15
0
∘
\angle PAQ > 150^\circ
∠
P
A
Q
>
15
0
∘
. (Consider the angle
P
A
Q
PAQ
P
A
Q
that does not contain side
A
B
AB
A
B
of the triangle.)Proposed by Alireza Dadgarnia
2019 IGO Advanced P5
Let points
A
,
B
A, B
A
,
B
and
C
C
C
lie on the parabola
Δ
\Delta
Δ
such that the point
H
H
H
, orthocenter of triangle
A
B
C
ABC
A
BC
, coincides with the focus of parabola
Δ
\Delta
Δ
. Prove that by changing the position of points
A
,
B
A, B
A
,
B
and
C
C
C
on
Δ
\Delta
Δ
so that the orthocenter remain at
H
H
H
, inradius of triangle
A
B
C
ABC
A
BC
remains unchanged.Proposed by Mahdi Etesamifard
4
3
Hide problems
2019 IGO Elementary P4
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is given such that
∠
D
A
C
=
∠
C
A
B
=
6
0
∘
,
\angle DAC = \angle CAB = 60^\circ,
∠
D
A
C
=
∠
C
A
B
=
6
0
∘
,
and
A
B
=
B
D
−
A
C
.
AB = BD - AC.
A
B
=
B
D
−
A
C
.
Lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect each other at point
E
E
E
. Prove that
∠
A
D
B
=
2
∠
B
E
C
.
\angle ADB = 2\angle BEC.
∠
A
D
B
=
2∠
BEC
.
Proposed by Iman Maghsoudi
2019 IGO Intermediate P4
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and let
K
K
K
be a point on line
A
D
AD
A
D
such that
B
K
=
A
B
BK=AB
B
K
=
A
B
. Suppose that
P
P
P
is an arbitrary point on
A
B
AB
A
B
, and the perpendicular bisector of
P
C
PC
PC
intersects the circumcircle of triangle
A
P
D
APD
A
P
D
at points
X
X
X
,
Y
Y
Y
. Prove that the circumcircle of triangle
A
B
K
ABK
A
B
K
passes through the orthocenter of triangle
A
X
Y
AXY
A
X
Y
.Proposed by Iman Maghsoudi
2019 IGO Advanced P4
Given an acute non-isosceles triangle
A
B
C
ABC
A
BC
with circumcircle
Γ
\Gamma
Γ
.
M
M
M
is the midpoint of segment
B
C
BC
BC
and
N
N
N
is the midpoint of arc
B
C
BC
BC
of
Γ
\Gamma
Γ
(the one that doesn't contain
A
A
A
).
X
X
X
and
Y
Y
Y
are points on
Γ
\Gamma
Γ
such that
B
X
∥
C
Y
∥
A
M
BX\parallel CY\parallel AM
BX
∥
C
Y
∥
A
M
. Assume there exists point
Z
Z
Z
on segment
B
C
BC
BC
such that circumcircle of triangle
X
Y
Z
XYZ
X
Y
Z
is tangent to
B
C
BC
BC
. Let
ω
\omega
ω
be the circumcircle of triangle
Z
M
N
ZMN
ZMN
. Line
A
M
AM
A
M
meets
ω
\omega
ω
for the second time at
P
P
P
. Let
K
K
K
be a point on
ω
\omega
ω
such that
K
N
∥
A
M
KN\parallel AM
K
N
∥
A
M
,
ω
b
\omega_b
ω
b
be a circle that passes through
B
B
B
,
X
X
X
and tangents to
B
C
BC
BC
and
ω
c
\omega_c
ω
c
be a circle that passes through
C
C
C
,
Y
Y
Y
and tangents to
B
C
BC
BC
. Prove that circle with center
K
K
K
and radius
K
P
KP
K
P
is tangent to 3 circles
ω
b
\omega_b
ω
b
,
ω
c
\omega_c
ω
c
and
Γ
\Gamma
Γ
.Proposed by Tran Quan - Vietnam
3
3
Hide problems
2019 IGO Elementary P3
There are
n
>
2
n>2
n
>
2
lines on the plane in general position; Meaning any two of them meet, but no three are concurrent. All their intersection points are marked, and then all the lines are removed, but the marked points are remained. It is not known which marked point belongs to which two lines. Is it possible to know which line belongs where, and restore them all?Proposed by Boris Frenkin - Russia
2019 IGO Intermediate P3
Three circles
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
and
ω
3
\omega_3
ω
3
pass through one common point, say
P
P
P
. The tangent line to
ω
1
\omega_1
ω
1
at
P
P
P
intersects
ω
2
\omega_2
ω
2
and
ω
3
\omega_3
ω
3
for the second time at points
P
1
,
2
P_{1,2}
P
1
,
2
and
P
1
,
3
P_{1,3}
P
1
,
3
, respectively. Points
P
2
,
1
P_{2,1}
P
2
,
1
,
P
2
,
3
P_{2,3}
P
2
,
3
,
P
3
,
1
P_{3,1}
P
3
,
1
and
P
3
,
2
P_{3,2}
P
3
,
2
are similarly defined. Prove that the perpendicular bisector of segments
P
1
,
2
P
1
,
3
P_{1,2}P_{1,3}
P
1
,
2
P
1
,
3
,
P
2
,
1
P
2
,
3
P_{2,1}P_{2,3}
P
2
,
1
P
2
,
3
and
P
3
,
1
P
3
,
2
P_{3,1}P_{3,2}
P
3
,
1
P
3
,
2
are concurrent.Proposed by Mahdi Etesamifard
2019 IGO Advanced P3
Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
have centres
O
1
O_1
O
1
and
O
2
O_2
O
2
, respectively. These two circles intersect at points
X
X
X
and
Y
Y
Y
.
A
B
AB
A
B
is common tangent line of these two circles such that
A
A
A
lies on
ω
1
\omega_1
ω
1
and
B
B
B
lies on
ω
2
\omega_2
ω
2
. Let tangents to
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
at
X
X
X
intersect
O
1
O
2
O_1O_2
O
1
O
2
at points
K
K
K
and
L
L
L
, respectively. Suppose that line
B
L
BL
B
L
intersects
ω
2
\omega_2
ω
2
for the second time at
M
M
M
and line
A
K
AK
A
K
intersects
ω
1
\omega_1
ω
1
for the second time at
N
N
N
. Prove that lines
A
M
,
B
N
AM, BN
A
M
,
BN
and
O
1
O
2
O_1O_2
O
1
O
2
concur. Proposed by Dominik Burek - Poland
2
3
Hide problems
2019 IGO Elementary P2
As shown in the figure, there are two rectangles
A
B
C
D
ABCD
A
BC
D
and
P
Q
R
D
PQRD
PQR
D
with the same area, and with parallel corresponding edges. Let points
N
,
N,
N
,
M
M
M
and
T
T
T
be the midpoints of segments
Q
R
,
QR,
QR
,
P
C
PC
PC
and
A
B
AB
A
B
, respectively. Prove that points
N
,
M
N,M
N
,
M
and
T
T
T
lie on the same line.http://s4.picofile.com/file/8372959484/E02.pngProposed by Morteza Saghafian
2019 IGO Intermediate P2
Find all quadrilaterals
A
B
C
D
ABCD
A
BC
D
such that all four triangles
D
A
B
DAB
D
A
B
,
C
D
A
CDA
C
D
A
,
B
C
D
BCD
BC
D
and
A
B
C
ABC
A
BC
are similar to one-another.Proposed by Morteza Saghafian
2019 IGO Advanced P2
Is it true that in any convex
n
n
n
-gon with
n
>
3
n > 3
n
>
3
, there exists a vertex and a diagonal passing through this vertex such that the angles of this diagonal with both sides adjacent to this vertex are acute?Proposed by Boris Frenkin - Russia