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Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2001 Iran MO (2nd round)
2001 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
3
2
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3 axises - Iran NMO 2001 (Second Round) - Problem3
Find all positive integers
n
n
n
such that we can put
n
n
n
equal squares on the plane that their sides are horizontal and vertical and the shape after putting the squares has at least
3
3
3
axises.
1*\infty table - Iran NMO 2001 (Second Round) - Problem6
Suppose a table with one row and infinite columns. We call each
1
×
1
1\times1
1
×
1
square a room. Let the table be finite from left. We number the rooms from left to
∞
\infty
∞
. We have put in some rooms some coins (A room can have more than one coin.). We can do
2
2
2
below operations:
a
)
a)
a
)
If in
2
2
2
adjacent rooms, there are some coins, we can move one coin from the left room
2
2
2
rooms to right and delete one room from the right room.
b
)
b)
b
)
If a room whose number is
3
3
3
or more has more than
1
1
1
coin, we can move one of its coins
1
1
1
room to right and move one other coin
2
2
2
rooms to left.
i
)
i)
i
)
Prove that for any initial configuration of the coins, after a finite number of movements, we cannot do anything more.
i
i
)
ii)
ii
)
Suppose that there is exactly one coin in each room from
1
1
1
to
n
n
n
. Prove that by doing the allowed operations, we cannot put any coins in the room
n
+
2
n+2
n
+
2
or the righter rooms.
2
2
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B'M perp to A'C' - Iran NMO 2001 (Second Round) - Problem2
Let
A
B
C
ABC
A
BC
be an acute triangle. We draw
3
3
3
triangles
B
′
A
C
,
C
′
A
B
,
A
′
B
C
B'AC,C'AB,A'BC
B
′
A
C
,
C
′
A
B
,
A
′
BC
on the sides of
Δ
A
B
C
\Delta ABC
Δ
A
BC
at the out sides such that:
∠
B
′
A
C
=
∠
C
′
B
A
=
∠
A
′
B
C
=
3
0
∘
,
∠
B
′
C
A
=
∠
C
′
A
B
=
∠
A
′
C
B
=
6
0
∘
\angle{B'AC}=\angle{C'BA}=\angle{A'BC}=30^{\circ} \ \ \ , \ \ \ \angle{B'CA}=\angle{C'AB}=\angle{A'CB}=60^{\circ}
∠
B
′
A
C
=
∠
C
′
B
A
=
∠
A
′
BC
=
3
0
∘
,
∠
B
′
C
A
=
∠
C
′
A
B
=
∠
A
′
CB
=
6
0
∘
If
M
M
M
is the midpoint of side
B
C
BC
BC
, prove that
B
′
M
B'M
B
′
M
is perpendicular to
A
′
C
′
A'C'
A
′
C
′
.
How much is angleA - Iran NMO 2001 (Second Round) - Problem5
In triangle
A
B
C
ABC
A
BC
,
A
B
>
A
C
AB>AC
A
B
>
A
C
. The bisectors of
∠
B
,
∠
C
\angle{B},\angle{C}
∠
B
,
∠
C
intersect the sides
A
C
,
A
B
AC,AB
A
C
,
A
B
at
P
,
Q
P,Q
P
,
Q
, respectively. Let
I
I
I
be the incenter of
Δ
A
B
C
\Delta ABC
Δ
A
BC
. Suppose that
I
P
=
I
Q
IP=IQ
I
P
=
I
Q
. How much isthe value of
∠
A
\angle A
∠
A
?
1
2
Hide problems
1+np - Iran NMO 2001 (Second Round) - Problem1
Let
n
n
n
be a positive integer and
p
p
p
be a prime number such that
n
p
+
1
np+1
n
p
+
1
is a perfect square. Prove that
n
+
1
n+1
n
+
1
can be written as the sum of
p
p
p
perfect squares.
P(x) Polynomial - Iran NMO 2001 (Second Round) - Problem4
Find all polynomials
P
P
P
with real coefficients such that
∀
x
∈
R
\forall{x\in\mathbb{R}}
∀
x
∈
R
we have:
P
(
2
P
(
x
)
)
=
2
P
(
P
(
x
)
)
+
2
(
P
(
x
)
)
2
.
P(2P(x))=2P(P(x))+2(P(x))^2.
P
(
2
P
(
x
))
=
2
P
(
P
(
x
))
+
2
(
P
(
x
)
)
2
.