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Problems
Contests
International Contests
Iranian Geometry Olympiad
2016 Iranian Geometry Olympiad
2016 Iranian Geometry Olympiad
Part of
Iranian Geometry Olympiad
Subcontests
(5)
5
3
Hide problems
AB = BC + AD in convex ABCD with many data given
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with these properties:
∠
A
D
C
=
13
5
o
\angle ADC = 135^o
∠
A
D
C
=
13
5
o
and
∠
A
D
B
−
∠
A
B
D
=
2
∠
D
A
B
=
4
∠
C
B
D
\angle ADB - \angle ABD = 2\angle DAB = 4\angle CBD
∠
A
D
B
−
∠
A
B
D
=
2∠
D
A
B
=
4∠
CB
D
. If
B
C
=
2
C
D
BC = \sqrt2 CD
BC
=
2
C
D
, prove that
A
B
=
B
C
+
A
D
AB = BC + AD
A
B
=
BC
+
A
D
.by Mahdi Etesami Fard
Perpendicular bisector of secant tangent to circumcircle
Let the circles
ω
\omega
ω
and
ω
′
\omega'
ω
′
intersect in points
A
A
A
and
B
B
B
. The tangent to circle
ω
\omega
ω
at
A
A
A
intersects
ω
′
\omega'
ω
′
at
C
C
C
and the tangent to circle
ω
′
\omega'
ω
′
at
A
A
A
intersects
ω
\omega
ω
at
D
D
D
. Suppose that the internal bisector of
∠
C
A
D
\angle CAD
∠
C
A
D
intersects
ω
\omega
ω
and
ω
′
\omega'
ω
′
at
E
E
E
and
F
F
F
, respectively, and the external bisector of
∠
C
A
D
\angle CAD
∠
C
A
D
intersects
ω
\omega
ω
and
ω
′
\omega'
ω
′
at
X
X
X
and
Y
Y
Y
, respectively. Prove that the perpendicular bisector of
X
Y
XY
X
Y
is tangent to the circumcircle of triangle
B
E
F
BEF
BEF
.Proposed by Mahdi Etesami Fard
Iranian Geometry Olympiad (5)
Do there exist six points
X
1
,
X
2
,
Y
1
,
Y
2
,
Z
1
,
Z
2
X_1,X_2,Y_1, Y_2,Z_1,Z_2
X
1
,
X
2
,
Y
1
,
Y
2
,
Z
1
,
Z
2
in the plane such that all of the triangles
X
i
Y
j
Z
k
X_iY_jZ_k
X
i
Y
j
Z
k
are similar for
1
≤
i
,
j
,
k
≤
2
1\leq i, j, k \leq 2
1
≤
i
,
j
,
k
≤
2
? Proposed by Morteza Saghafian
4
3
Hide problems
angles in right triangle, with perpendicular bisectors
In a right-angled triangle
A
B
C
ABC
A
BC
(
∠
A
=
9
0
o
\angle A = 90^o
∠
A
=
9
0
o
), the perpendicular bisector of
B
C
BC
BC
intersects the line
A
C
AC
A
C
in
K
K
K
and the perpendicular bisector of
B
K
BK
B
K
intersects the line
A
B
AB
A
B
in
L
L
L
. If the line
C
L
CL
C
L
be the internal bisector of angle
C
C
C
, find all possible values for angles
B
B
B
and
C
C
C
.by Mahdi Etesami Fard
Similarity through arc midpoint in right triangle
Let
ω
\omega
ω
be the circumcircle of right-angled triangle
A
B
C
ABC
A
BC
(
∠
A
=
9
0
∘
\angle A = 90^{\circ}
∠
A
=
9
0
∘
). The tangent to
ω
\omega
ω
at point
A
A
A
intersects the line
B
C
BC
BC
at point
P
P
P
. Suppose that
M
M
M
is the midpoint of the minor arc
A
B
AB
A
B
, and
P
M
PM
PM
intersects
ω
\omega
ω
for the second time in
Q
Q
Q
. The tangent to
ω
\omega
ω
at point
Q
Q
Q
intersects
A
C
AC
A
C
at
K
K
K
. Prove that
∠
P
K
C
=
9
0
∘
\angle PKC = 90^{\circ}
∠
P
K
C
=
9
0
∘
.Proposed by Davood Vakili
Iranian Geometry Olympiad (4)
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, the lines
A
B
AB
A
B
and
C
D
CD
C
D
meet at point
E
E
E
and the lines
A
D
AD
A
D
and
B
C
BC
BC
meet at point
F
F
F
. Let
P
P
P
be the intersection point of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. Suppose that
ω
1
\omega_1
ω
1
is a circle passing through
D
D
D
and tangent to
A
C
AC
A
C
at
P
P
P
. Also suppose that
ω
2
\omega_2
ω
2
is a circle passing through
C
C
C
and tangent to
B
D
BD
B
D
at
P
P
P
. Let
X
X
X
be the intersection point of
ω
1
\omega_1
ω
1
and
A
D
AD
A
D
, and
Y
Y
Y
be the intersection point of
ω
2
\omega_2
ω
2
and
B
C
BC
BC
. Suppose that the circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
intersect each other in
Q
Q
Q
for the second time. Prove that the perpendicular from
P
P
P
to the line
E
F
EF
EF
passes through the circumcenter of triangle
X
Q
Y
XQY
XQ
Y
. Proposed by Iman Maghsoudi
3
3
Hide problems
Dissect triangle to N similar quadrilaterals
Find all positive integers
N
N
N
such that there exists a triangle which can be dissected into
N
N
N
similar quadrilaterals.Proposed by Nikolai Beluhov (Bulgaria) and Morteza Saghafian
2 parallelograms on consecutive sides of ABCD
Suppose that
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral with no parallel sides. Make a parallelogram on each two consecutive sides. Show that among these
4
4
4
new points, there is only one point inside the quadrilateral
A
B
C
D
ABCD
A
BC
D
.by Morteza Saghafian
Iranian Geometry Olympiad (3)
In a convex qualrilateral
A
B
C
D
ABCD
A
BC
D
, let
P
P
P
be the intersection point of
A
D
AD
A
D
and
B
C
BC
BC
. Suppose that
I
1
I_1
I
1
and
I
2
I_2
I
2
are the incenters of triangles
P
A
B
PAB
P
A
B
and
P
D
C
PDC
P
D
C
,respectively. Let
O
O
O
be the circumcenter of
P
A
B
PAB
P
A
B
, and
H
H
H
the orthocenter of
P
D
C
PDC
P
D
C
. Show that the circumcircles of triangles
A
I
1
B
AI_1B
A
I
1
B
and
D
H
C
DHC
DH
C
are tangent together if and only if the circumcircles of triangles
A
O
B
AOB
A
OB
and
D
I
2
C
DI_2C
D
I
2
C
are tangent together. Proposed by Hooman Fattahimoghaddam
2
3
Hide problems
equal segments concerning circumcircle
Let
ω
\omega
ω
be the circumcircle of triangle
A
B
C
ABC
A
BC
with
A
C
>
A
B
AC > AB
A
C
>
A
B
. Let
X
X
X
be a point on
A
C
AC
A
C
and
Y
Y
Y
be a point on the circle
ω
\omega
ω
, such that
C
X
=
C
Y
=
A
B
CX = CY = AB
CX
=
C
Y
=
A
B
. (The points
A
A
A
and
Y
Y
Y
lie on different sides of the line
B
C
BC
BC
). The line
X
Y
XY
X
Y
intersects
ω
\omega
ω
for the second time in point
P
P
P
. Show that
P
B
=
P
C
PB = PC
PB
=
PC
.by Iman Maghsoudi
Tangent intersect intersect tangent intersect
Let two circles
C
1
C_1
C
1
and
C
2
C_2
C
2
intersect in points
A
A
A
and
B
B
B
. The tangent to
C
1
C_1
C
1
at
A
A
A
intersects
C
2
C_2
C
2
in
P
P
P
and the line
P
B
PB
PB
intersects
C
1
C_1
C
1
for the second time in
Q
Q
Q
(suppose that
Q
Q
Q
is outside
C
2
C_2
C
2
). The tangent to
C
2
C_2
C
2
from
Q
Q
Q
intersects
C
1
C_1
C
1
and
C
2
C_2
C
2
in
C
C
C
and
D
D
D
, respectively. (The points
A
A
A
and
D
D
D
lie on different sides of the line
P
Q
PQ
PQ
.) Show that
A
D
AD
A
D
is the bisector of
∠
C
A
P
\angle CAP
∠
C
A
P
.Proposed by Iman Maghsoudi
Iranian Geometry Olympiad (2)
In acute-angled triangle
A
B
C
ABC
A
BC
, altitude of
A
A
A
meets
B
C
BC
BC
at
D
D
D
, and
M
M
M
is midpoint of
A
C
AC
A
C
. Suppose that
X
X
X
is a point such that
∡
A
X
B
=
∡
D
X
M
=
9
0
∘
\measuredangle AXB = \measuredangle DXM =90^\circ
∡
A
XB
=
∡
D
XM
=
9
0
∘
(assume that
X
X
X
and
C
C
C
lie on opposite sides of the line
B
M
BM
BM
). Show that
∡
X
M
B
=
2
∡
M
B
C
\measuredangle XMB = 2\measuredangle MBC
∡
XMB
=
2
∡
MBC
.Proposed by Davood Vakili
1
3
Hide problems
shortest path in a grid
Ali wants to move from point
A
A
A
to point
B
B
B
. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between
A
A
A
and
B
B
B
. Only draw the path and write its length. https://1.bp.blogspot.com/-nZrxJLfIAp8/W1RyCdnhl3I/AAAAAAAAIzQ/NM3t5EtJWMcWQS0ig0IghSo54DQUBH5hwCK4BGAYYCw/s1600/igo%2B2016.el1.png by Morteza Saghafian
Secant less than half trapezoid perimeter
In trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
∣
∣
C
D
AB || CD
A
B
∣∣
C
D
,
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
are two circles with diameters
A
D
AD
A
D
and
B
C
BC
BC
, respectively. Let
X
X
X
and
Y
Y
Y
be two arbitrary points on
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
, respectively. Show that the length of segment
X
Y
XY
X
Y
is not more than half the perimeter of
A
B
C
D
ABCD
A
BC
D
.Proposed by Mahdi Etesami Fard
3 points are collinear!
Let the circles
ω
\omega
ω
and
ω
′
\omega^ \prime
ω
′
intersect in
A
A
A
and
B
B
B
. Tangent to circle
ω
\omega
ω
at
A
A
A
intersects
ω
′
\omega^ \prime
ω
′
in
C
C
C
and tangent to circle
ω
′
\omega^ \prime
ω
′
at
A
A
A
intersects
ω
\omega
ω
in
D
D
D
. Suppose that
C
D
CD
C
D
intersects
ω
\omega
ω
and
ω
′
\omega^ \prime
ω
′
in
E
E
E
and
F
F
F
, respectively (assume that
E
E
E
is between
F
F
F
and
C
C
C
). The perpendicular to
A
C
AC
A
C
from
E
E
E
intersects
ω
′
\omega^ \prime
ω
′
in point
P
P
P
and perpendicular to
A
D
AD
A
D
from
F
F
F
intersects
ω
\omega
ω
in point
Q
Q
Q
(The points
A
,
P
A, P
A
,
P
and
Q
Q
Q
lie on the same side of the line
C
D
CD
C
D
). Prove that the points
A
,
P
A, P
A
,
P
and
Q
Q
Q
are collinear. Proposed by Mahdi Etesami Fard