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Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2005 Iran MO (2nd round)
2005 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(3)
1
2
Hide problems
There exists a finite configuratio- Iran NMO 2005 - Problem4
We have a
2
×
n
2\times n
2
×
n
rectangle. We call each
1
×
1
1\times1
1
×
1
square a room and we show the room in the
i
t
h
i^{th}
i
t
h
row and
j
t
h
j^{th}
j
t
h
column as
(
i
,
j
)
(i,j)
(
i
,
j
)
. There are some coins in some rooms of the rectangle. If there exist more than
1
1
1
coin in each room, we can delete
2
2
2
coins from it and add
1
1
1
coin to its right adjacent room OR we can delete
2
2
2
coins from it and add
1
1
1
coin to its up adjacent room. Prove that there exists a finite configuration of allowable operations such that we can put a coin in the room
(
1
,
n
)
(1,n)
(
1
,
n
)
.
Prove that 4p-3 is a square - Iran NMO 2005 - Problem1
Let
n
,
p
>
1
n,p>1
n
,
p
>
1
be positive integers and
p
p
p
be prime. We know that
n
∣
p
−
1
n|p-1
n
∣
p
−
1
and
p
∣
n
3
−
1
p|n^3-1
p
∣
n
3
−
1
. Prove that
4
p
−
3
4p-3
4
p
−
3
is a perfect square.
2
2
Hide problems
PB is perpendicular to MK - Iran NMO 2005 - Problem5
B
C
BC
BC
is a diameter of a circle and the points
X
,
Y
X,Y
X
,
Y
are on the circle such that
X
Y
⊥
B
C
XY\perp BC
X
Y
⊥
BC
. The points
P
,
M
P,M
P
,
M
are on
X
Y
,
C
Y
XY,CY
X
Y
,
C
Y
(or their stretches), respectively, such that
C
Y
∣
∣
P
B
CY||PB
C
Y
∣∣
PB
and
C
X
∣
∣
P
M
CX||PM
CX
∣∣
PM
. Let
K
K
K
be the meet point of the lines
X
C
,
B
P
XC,BP
XC
,
BP
. Prove that
P
B
⊥
M
K
PB\perp MK
PB
⊥
M
K
.
MN passes through a constant point- Iran NMO 2005 - Problem2
In triangle
A
B
C
ABC
A
BC
,
∠
A
=
6
0
∘
\angle A=60^{\circ}
∠
A
=
6
0
∘
. The point
D
D
D
changes on the segment
B
C
BC
BC
. Let
O
1
,
O
2
O_1,O_2
O
1
,
O
2
be the circumcenters of the triangles
Δ
A
B
D
,
Δ
A
C
D
\Delta ABD,\Delta ACD
Δ
A
B
D
,
Δ
A
C
D
, respectively. Let
M
M
M
be the meet point of
B
O
1
,
C
O
2
BO_1,CO_2
B
O
1
,
C
O
2
and let
N
N
N
be the circumcenter of
Δ
D
O
1
O
2
\Delta DO_1O_2
Δ
D
O
1
O
2
. Prove that, by changing
D
D
D
on
B
C
BC
BC
, the line
M
N
MN
MN
passes through a constant point.
3
2
Hide problems
M has at least 79 members - Iran NMO 2005 - Problem3
In one galaxy, there exist more than one million stars. Let
M
M
M
be the set of the distances between any
2
2
2
of them. Prove that, in every moment,
M
M
M
has at least
79
79
79
members. (Suppose each star as a point.)
Find all functions (x+y)f(f(x)y)=x^2f(f(x)+f(y)) - Iran NMO 2005 - Problem6
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R}^{+}\to \mathbb{R}^{+}
f
:
R
+
→
R
+
such that for all positive real numbers
x
x
x
and
y
y
y
, the following equation holds:
(
x
+
y
)
f
(
f
(
x
)
y
)
=
x
2
f
(
f
(
x
)
+
f
(
y
)
)
.
(x+y)f(f(x)y)=x^2f(f(x)+f(y)).
(
x
+
y
)
f
(
f
(
x
)
y
)
=
x
2
f
(
f
(
x
)
+
f
(
y
))
.