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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2007 Iran MO (2nd Round)
2007 Iran MO (2nd Round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
2
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Buildings in a city - Iran NMO 2007 - Problem3
In a city, there are some buildings. We say the building
A
A
A
is dominant to the building
B
B
B
if the line that connects upside of
A
A
A
to upside of
B
B
B
makes an angle more than
4
5
∘
45^{\circ}
4
5
∘
with earth. We want to make a building in a given location. Suppose none of the buildings are dominant to each other. Prove that we can make the building with a height such that again, none of the buildings are dominant to each other. (Suppose the city as a horizontal plain and each building as a perpendicular line to the plain.)
Printer doesn't print 13824 - Iran NMO 2007 - Problem6
Farhad has made a machine. When the machine starts, it prints some special numbers. The property of this machine is that for every positive integer
n
n
n
, it prints exactly one of the numbers
n
,
2
n
,
3
n
n,2n,3n
n
,
2
n
,
3
n
. We know that the machine prints
2
2
2
. Prove that it doesn't print
13824
13824
13824
.
1
2
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Iran NMO 2007 (Second Round) - Problem1
In triangle
A
B
C
ABC
A
BC
,
∠
A
=
9
0
∘
\angle A=90^{\circ}
∠
A
=
9
0
∘
and
M
M
M
is the midpoint of
B
C
BC
BC
. Point
D
D
D
is chosen on segment
A
C
AC
A
C
such that
A
M
=
A
D
AM=AD
A
M
=
A
D
and
P
P
P
is the second meet point of the circumcircles of triangles
Δ
A
M
C
,
Δ
B
D
C
\Delta AMC,\Delta BDC
Δ
A
MC
,
Δ
B
D
C
. Prove that the line
C
P
CP
CP
bisects
∠
A
C
B
\angle ACB
∠
A
CB
.
Iran NMO 2007 (Second Round) - Problem4
Prove that for every positive integer
n
n
n
, there exist
n
n
n
positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.
2
2
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Prove that PE/PF=ME/NF - Iran NMO 2007 - Problem5
Tow circles
C
,
D
C,D
C
,
D
are exterior tangent to each other at point
P
P
P
. Point
A
A
A
is in the circle
C
C
C
. We draw
2
2
2
tangents
A
M
,
A
N
AM,AN
A
M
,
A
N
from
A
A
A
to the circle
D
D
D
(
M
,
N
M,N
M
,
N
are the tangency points.). The second meet points of
A
M
,
A
N
AM,AN
A
M
,
A
N
with
C
C
C
are
E
,
F
E,F
E
,
F
, respectively. Prove that
P
E
P
F
=
M
E
N
F
\frac{PE}{PF}=\frac{ME}{NF}
PF
PE
=
NF
ME
.
Vertices of a cube - Iran NMO 2007 - Problem2
Two vertices of a cube are
A
,
O
A,O
A
,
O
such that
A
O
AO
A
O
is the diagonal of one its faces. A
n
−
n-
n
−
run is a sequence of
n
+
1
n+1
n
+
1
vertices of the cube such that each
2
2
2
consecutive vertices in the sequence are
2
2
2
ends of one side of the cube. Is the
1386
−
1386-
1386
−
runs from
O
O
O
to itself less than
1386
−
1386-
1386
−
runs from
O
O
O
to
A
A
A
or more than it?