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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2004 Iran MO (2nd round)
2004 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(6)
4
1
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f(m)+f(n)|m+n - Iran NMO 2004 (Second Round) - Problem4
N
\mathbb{N}
N
is the set of positive integers. Determine all functions
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
such that for every pair
(
m
,
n
)
∈
N
2
(m,n)\in\mathbb{N}^2
(
m
,
n
)
∈
N
2
we have that:
f
(
m
)
+
f
(
n
)
∣
m
+
n
.
f(m)+f(n) \ | \ m+n .
f
(
m
)
+
f
(
n
)
∣
m
+
n
.
6
1
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Alligator Coin - Iran NMO 2004 (Second Round) - Problem6
We have a
m
×
n
m\times n
m
×
n
table and
m
≥
4
m\geq{4}
m
≥
4
and we call a
1
×
1
1\times 1
1
×
1
square a room. When we put an alligator coin in a room, it menaces all the rooms in his column and his adjacent rooms in his row. What's the minimum number of alligator coins required, such that each room is menaced at least by one alligator coin? (Notice that all alligator coins are vertical.)
5
1
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bisector of XAY - Iran NMO 2004 (Second Round) - Problem5
The interior bisector of
∠
A
\angle A
∠
A
from
△
A
B
C
\triangle ABC
△
A
BC
intersects the side
B
C
BC
BC
and the circumcircle of
Δ
A
B
C
\Delta ABC
Δ
A
BC
at
D
,
M
D,M
D
,
M
, respectively. Let
ω
\omega
ω
be a circle with center
M
M
M
and radius
M
B
MB
MB
. A line passing through
D
D
D
, intersects
ω
\omega
ω
at
X
,
Y
X,Y
X
,
Y
. Prove that
A
D
AD
A
D
bisects
∠
X
A
Y
\angle XAY
∠
X
A
Y
.
3
1
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Road Ministry - Iran NMO 2004 (Second Round) - Problem3
The road ministry has assigned
80
80
80
informal companies to repair
2400
2400
2400
roads. These roads connect
100
100
100
cities to each other. Each road is between
2
2
2
cities and there is at most
1
1
1
road between every
2
2
2
cities. We know that each company repairs
30
30
30
roads that it has agencies in each
2
2
2
ends of them. Prove that there exists a city in which
8
8
8
companies have agencies.
1
1
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AD/DI_a - Iran NMO 2004 (Second Round) - Problem1
A
B
C
ABC
A
BC
is a triangle and
∠
A
=
9
0
∘
\angle A=90^{\circ}
∠
A
=
9
0
∘
. Let
D
D
D
be the meet point of the interior bisector of
∠
A
\angle A
∠
A
and
B
C
BC
BC
. And let
I
a
I_a
I
a
be the
A
−
A-
A
−
excenter of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that:
A
D
D
I
a
≤
2
−
1.
\frac{AD}{DI_a}\leq\sqrt{2}-1.
D
I
a
A
D
≤
2
−
1.
2
1
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Ascendant Function - Iran NMO 2004 (Second Round) - Problem2
Let
f
:
R
≥
0
→
R
f:\mathbb{R}^{\geq 0}\to\mathbb{R}
f
:
R
≥
0
→
R
be a function such that
f
(
x
)
−
3
x
f(x)-3x
f
(
x
)
−
3
x
and
f
(
x
)
−
x
3
f(x)-x^3
f
(
x
)
−
x
3
are ascendant functions. Prove that
f
(
x
)
−
x
2
−
x
f(x)-x^2-x
f
(
x
)
−
x
2
−
x
is an ascendant function, too. (We call the function
g
(
x
)
g(x)
g
(
x
)
ascendant, when for every
x
≤
y
x\leq{y}
x
≤
y
we have
g
(
x
)
≤
g
(
y
)
g(x)\leq{g(y)}
g
(
x
)
≤
g
(
y
)
.)