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Problems
Contests
International Contests
Iranian Geometry Olympiad
2015 Iran Geometry Olympiad
2015 Iran Geometry Olympiad
Part of
Iranian Geometry Olympiad
Subcontests
(5)
4
2
Hide problems
4 triangles with equal areas, prove MNPQ parallelogram
In rectangle
A
B
C
D
ABCD
A
BC
D
, the points
M
,
N
,
P
,
Q
M,N,P, Q
M
,
N
,
P
,
Q
lie on
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
respectively such that the area of triangles
A
Q
M
AQM
A
QM
,
B
M
N
BMN
BMN
,
C
N
P
CNP
CNP
,
D
P
Q
DPQ
D
PQ
are equal. Prove that the quadrilateral
M
N
P
Q
MNPQ
MNPQ
is parallelogram.by Mahdi Etesami Fard
Prove isoceles triangle given 12 points on 2 circles
In triangle
A
B
C
ABC
A
BC
, we draw the circle with center
A
A
A
and radius
A
B
AB
A
B
. This circle intersects
A
C
AC
A
C
at two points. Also we draw the circle with center
A
A
A
and radius
A
C
AC
A
C
and this circle intersects
A
B
AB
A
B
at two points. Denote these four points by
A
1
,
A
2
,
A
3
,
A
4
A_1, A_2, A_3, A_4
A
1
,
A
2
,
A
3
,
A
4
. Find the points
B
1
,
B
2
,
B
3
,
B
4
B_1, B_2, B_3, B_4
B
1
,
B
2
,
B
3
,
B
4
and
C
1
,
C
2
,
C
3
,
C
4
C_1, C_2, C_3, C_4
C
1
,
C
2
,
C
3
,
C
4
similarly. Suppose that these
12
12
12
points lie on two circles. Prove that the triangle
A
B
C
ABC
A
BC
is isosceles.
5
3
Hide problems
6 circles in plane such every circle passes through centers of 3 others
Do there exist
6
6
6
circles in the plane such that every circle passes through centers of exactly
3
3
3
other circles?by Morteza Saghafian
Geometry, circles
a) Do there exist 5 circles in the plane such that every circle passes through centers of exactly 3 circles? b) Do there exist 6 circles in the plane such that every circle passes through centers of exactly 3 circles?
IGO - Hard level - Problem 5
we have a triangle
A
B
C
ABC
A
BC
and make rectangles
A
B
A
1
B
2
ABA_1B_2
A
B
A
1
B
2
,
B
C
B
1
C
2
BCB_1C_2
BC
B
1
C
2
and
C
A
C
1
A
2
CAC_1A_2
C
A
C
1
A
2
out of it.then pass a line through
A
2
A_2
A
2
perpendicular to
C
1
A
2
C_1A_2
C
1
A
2
and pass another line through
A
1
A_1
A
1
perpendicular to
A
1
B
2
A_1B_2
A
1
B
2
.let
A
′
A'
A
′
the common point of this two lines. like this we make
B
′
B'
B
′
and
C
′
C'
C
′
.prove
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
intersect each other in a same point.
3
3
Hide problems
ABC=90^o, BCD=30^o, AB = CD, BC = 2AD, prove BAD=30^o
In the figure below, we know that
A
B
=
C
D
AB = CD
A
B
=
C
D
and
B
C
=
2
A
D
BC = 2AD
BC
=
2
A
D
. Prove that
∠
B
A
D
=
3
0
o
\angle BAD = 30^o
∠
B
A
D
=
3
0
o
. https://3.bp.blogspot.com/-IXi_8jSwzlU/W1R5IydV5uI/AAAAAAAAIzo/2sREnDEnLH8R9zmAZLCkVCGeMaeITX9YwCK4BGAYYCw/s400/IGO%2B2015.el3.png
line AZ is perpendicular to BC
In triangle
A
B
C
ABC
A
BC
,
M
,
N
,
K
M,N,K
M
,
N
,
K
are midpoints of sides
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
,respectively.Construct two semicircles with diameter
A
B
,
A
C
AB,AC
A
B
,
A
C
outside of triangle
A
B
C
ABC
A
BC
.
M
K
,
M
N
MK,MN
M
K
,
MN
intersect with semicircles in
X
,
Y
X,Y
X
,
Y
.The tangents to semicircles at
X
,
Y
X,Y
X
,
Y
intersect at point
Z
Z
Z
.Prove that
A
Z
⊥
B
C
AZ \perp BC
A
Z
⊥
BC
.(Mehdi E'tesami Fard)
IGO - Hard level - Problem 3
let
H
H
H
the orthocenter of the triangle
A
B
C
ABC
A
BC
pass two lines
l
1
l_1
l
1
and
l
2
l_2
l
2
through
H
H
H
such that
l
1
⊥
l
2
l_1 \bot l_2
l
1
⊥
l
2
we have
l
1
∩
B
C
=
D
l_1 \cap BC = D
l
1
∩
BC
=
D
and
l
1
∩
A
B
=
Z
l_1 \cap AB = Z
l
1
∩
A
B
=
Z
also
l
2
∩
B
C
=
E
l_2 \cap BC = E
l
2
∩
BC
=
E
and
l
2
∩
A
C
=
X
l_2 \cap AC = X
l
2
∩
A
C
=
X
like this picturepass a line
d
1
d_1
d
1
through
D
D
D
parallel to
A
C
AC
A
C
and another line
d
2
d_2
d
2
through
E
E
E
parallel to
A
B
AB
A
B
let
d
1
∩
d
2
=
Y
d_1 \cap d_2 = Y
d
1
∩
d
2
=
Y
prove
X
X
X
,
,
,
Y
Y
Y
and
Z
Z
Z
are on a same line
2
3
Hide problems
find angles in triangle with A=60^o
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
6
0
o
\angle A = 60^o
∠
A
=
6
0
o
. The points
M
,
N
,
K
M,N,K
M
,
N
,
K
lie on
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
respectively such that
B
K
=
K
M
=
M
N
=
N
C
BK = KM = MN = NC
B
K
=
K
M
=
MN
=
NC
. If
A
N
=
2
A
K
AN = 2AK
A
N
=
2
A
K
, find the values of
∠
B
\angle B
∠
B
and
∠
C
\angle C
∠
C
.by Mahdi Etesami Fard
line passes through the circumenter
In acute-angled triangle
A
B
C
ABC
A
BC
,
B
H
BH
B
H
is the altitude of the vertex
B
B
B
. The points
D
D
D
and
E
E
E
are midpoints of
A
B
AB
A
B
and
A
C
AC
A
C
respectively. Suppose that
F
F
F
be the reflection of
H
H
H
with respect to
E
D
ED
E
D
. Prove that the line
B
F
BF
BF
passes through circumcenter of
A
B
C
ABC
A
BC
. by Davood Vakili
IGO - Hard level - Problem 2
let
A
B
C
ABC
A
BC
an equilateral triangle with circum circle
w
w
w
let
P
P
P
a point on arc
B
C
BC
BC
( point
A
A
A
is on the other side )pass a tangent line
d
d
d
through point
P
P
P
such that
P
∩
A
B
=
F
P \cap AB = F
P
∩
A
B
=
F
and
A
C
∩
d
=
L
AC \cap d = L
A
C
∩
d
=
L
let
O
O
O
the center of the circle
w
w
w
prove that
∠
L
O
F
>
9
0
0
\angle LOF > 90^{0}
∠
L
OF
>
9
0
0
1
3
Hide problems
convex polygons from 4 wooden triangles with sides 3, 4, 5
We have four wooden triangles with sides
3
,
4
,
5
3, 4, 5
3
,
4
,
5
centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof)A convex polygon is a polygon which all of it's angles are less than
18
0
o
180^o
18
0
o
and there isn't any hole in it. For example: https://1.bp.blogspot.com/-JgvF_B-uRag/W1R4f4AXxTI/AAAAAAAAIzc/Fo3qu3pxXcoElk01RTYJYZNwj0plJaKPQCK4BGAYYCw/s640/igo%2B2015.el1.png
IGO-Intermediate-Problem 1
Given a circle and Points
P
,
B
,
A
P,B,A
P
,
B
,
A
on it.Point
Q
Q
Q
is Interior of this circle such that:
1
)
1)
1
)
∠
P
A
Q
=
90
\angle PAQ=90
∠
P
A
Q
=
90
.
2
)
P
Q
=
B
Q
2)PQ=BQ
2
)
PQ
=
BQ
. Prove that
∠
A
Q
B
−
∠
P
Q
A
=
A
B
⌢
\angle AQB - \angle PQA=\stackrel{\frown}{AB}
∠
A
QB
−
∠
PQ
A
=
A
B
⌢
.proposed by Davoud Vakili, Iran.
IGO - Hard level - problem 1
let
w
1
w_1
w
1
and
w
2
w_2
w
2
two circles such that
w
1
∩
w
2
=
{
A
,
B
}
w_1 \cap w_2 = \{ A , B \}
w
1
∩
w
2
=
{
A
,
B
}
let
X
X
X
a point on
w
2
w_2
w
2
and
Y
Y
Y
on
w
1
w_1
w
1
such that
B
Y
⊥
B
X
BY \bot BX
B
Y
⊥
BX
suppose that
O
O
O
is the center of
w
1
w_1
w
1
and
X
′
=
w
2
∩
O
X
X' = w_2 \cap OX
X
′
=
w
2
∩
OX
now if
K
=
w
2
∩
X
′
Y
K = w_2 \cap X'Y
K
=
w
2
∩
X
′
Y
prove
X
X
X
is the midpoint of arc
A
K
AK
A
K