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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2003 Iran MO (2nd round)
2003 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
2
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Volleyball teams - Iran NMO 2003 (Second Round) - Problem3
n
n
n
volleyball teams have competed to each other (each
2
2
2
teams have competed exactly
1
1
1
time.). For every
2
2
2
distinct teams like
A
,
B
A,B
A
,
B
, there exist exactly
t
t
t
teams which have lost their match with
A
,
B
A,B
A
,
B
. Prove that
n
=
4
t
+
3
n=4t+3
n
=
4
t
+
3
. (Notabene that in volleyball, there doesn’t exist tie!)
Chessboard & Robot - Iran NMO 2003 (Second Round) - Problem6
We have a chessboard and we call a
1
×
1
1\times1
1
×
1
square a room. A robot is standing on one arbitrary vertex of the rooms. The robot starts to move and in every one movement, he moves one side of a room. This robot has
2
2
2
memories
A
,
B
A,B
A
,
B
. At first, the values of
A
,
B
A,B
A
,
B
are
0
0
0
. In each movement, if he goes up,
1
1
1
unit is added to
A
A
A
, and if he goes down,
1
1
1
unit is waned from
A
A
A
, and if he goes right, the value of
A
A
A
is added to
B
B
B
, and if he goes left, the value of
A
A
A
is waned from
B
B
B
. Suppose that the robot has traversed a traverse (!) which hasn’t intersected itself and finally, he has come back to its initial vertex. If
v
(
B
)
v(B)
v
(
B
)
is the value of
B
B
B
in the last of the traverse, prove that in this traverse, the interior surface of the shape that the robot has moved on its circumference is equal to
∣
v
(
B
)
∣
|v(B)|
∣
v
(
B
)
∣
.
2
2
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n house in village - Iran NMO 2003 (Second Round) - Problem2
In a village, there are
n
n
n
houses with
n
>
2
n>2
n
>
2
and all of them are not collinear. We want to generate a water resource in the village. For doing this, point
A
A
A
is better than point
B
B
B
if the sum of the distances from point
A
A
A
to the houses is less than the sum of the distances from point
B
B
B
to the houses. We call a point ideal if there doesn’t exist any better point than it. Prove that there exist at most
1
1
1
ideal point to generate the resource.
BT+CT <= 2R - Iran NMO 2003 (Second Round) - Problem5
∠
A
\angle{A}
∠
A
is the least angle in
Δ
A
B
C
\Delta{ABC}
Δ
A
BC
. Point
D
D
D
is on the arc
B
C
BC
BC
from the circumcircle of
Δ
A
B
C
\Delta{ABC}
Δ
A
BC
. The perpendicular bisectors of the segments
A
B
,
A
C
AB,AC
A
B
,
A
C
intersect the line
A
D
AD
A
D
at
M
,
N
M,N
M
,
N
, respectively. Point
T
T
T
is the meet point of
B
M
,
C
N
BM,CN
BM
,
CN
. Suppose that
R
R
R
is the radius of the circumcircle of
Δ
A
B
C
\Delta{ABC}
Δ
A
BC
. Prove that:
B
T
+
C
T
≤
2
R
.
BT+CT\leq{2R}.
BT
+
CT
≤
2
R
.
1
2
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3-stratum numbers - Iran NMO 2003 (Second Round) - Problem1
We call the positive integer
n
n
n
a
3
−
3-
3
−
stratum number if we can divide the set of its positive divisors into
3
3
3
subsets such that the sum of each subset is equal to the others.
a
)
a)
a
)
Find a
3
−
3-
3
−
stratum number.
b
)
b)
b
)
Prove that there are infinitely many
3
−
3-
3
−
stratum numbers.
Easy Inequality - Iran NMO 2003 (Second Round) - Problem4
Let
x
,
y
,
z
∈
R
x,y,z\in\mathbb{R}
x
,
y
,
z
∈
R
and
x
y
z
=
−
1
xyz=-1
x
yz
=
−
1
. Prove that:
x
4
+
y
4
+
z
4
+
3
(
x
+
y
+
z
)
≥
x
2
y
+
x
2
z
+
y
2
x
+
y
2
z
+
z
2
x
+
z
2
y
.
x^4+y^4+z^4+3(x+y+z)\geq\frac{x^2}{y}+\frac{x^2}{z}+\frac{y^2}{x}+\frac{y^2}{z}+\frac{z^2}{x}+\frac{z^2}{y}.
x
4
+
y
4
+
z
4
+
3
(
x
+
y
+
z
)
≥
y
x
2
+
z
x
2
+
x
y
2
+
z
y
2
+
x
z
2
+
y
z
2
.