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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2010 Iran MO (2nd Round)
2010 Iran MO (2nd Round)
Part of
Iran MO (2nd Round)
Subcontests
(6)
6
1
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Iran National Math Olympiad-Day 2-Problem 6
A school has
n
n
n
students and some super classes are provided for them. Each student can participate in any number of classes that he/she wants. Every class has at least two students participating in it. We know that if two different classes have at least two common students, then the number of the students in the first of these two classes is different from the number of the students in the second one. Prove that the number of classes is not greater that
(
n
−
1
)
2
\left(n-1\right)^2
(
n
−
1
)
2
.
5
1
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Iran National Math Olympiad-Day 2-Problem 5
In triangle
A
B
C
ABC
A
BC
we havev
∠
A
=
π
3
\angle A=\frac{\pi}{3}
∠
A
=
3
π
. Construct
E
E
E
and
F
F
F
on continue of
A
B
AB
A
B
and
A
C
AC
A
C
respectively such that
B
E
=
C
F
=
B
C
BE=CF=BC
BE
=
CF
=
BC
. Suppose that
E
F
EF
EF
meets circumcircle of
△
A
C
E
\triangle ACE
△
A
CE
in
K
K
K
. (
K
≢
E
K\not \equiv E
K
≡
E
). Prove that
K
K
K
is on the bisector of
∠
A
\angle A
∠
A
.
3
1
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Iran National Math Olympiad-Day 1-Problem 3
Circles
W
1
,
W
2
W_1,W_2
W
1
,
W
2
meet at
D
D
D
and
P
P
P
.
A
A
A
and
B
B
B
are on
W
1
,
W
2
W_1,W_2
W
1
,
W
2
respectively, such that
A
B
AB
A
B
is tangent to
W
1
W_1
W
1
and
W
2
W_2
W
2
. Suppose
D
D
D
is closer than
P
P
P
to the line
A
B
AB
A
B
.
A
D
AD
A
D
meet circle
W
2
W_2
W
2
for second time at
C
C
C
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Prove that
∠
D
P
M
=
∠
B
D
C
\angle{DPM}=\angle{BDC}
∠
D
PM
=
∠
B
D
C
.
2
1
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Iran National Math Olympiad-Day 1-Problem 2
There are
n
n
n
points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these
n
n
n
points and area of them is 1,is not greater than
2
3
(
n
2
−
n
)
\frac23(n^2-n)
3
2
(
n
2
−
n
)
.
1
1
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Iran National Math Olympiad-Day 1-Problem 1
Let
a
,
b
a,b
a
,
b
be two positive integers and
a
>
b
a>b
a
>
b
.We know that
gcd
(
a
−
b
,
a
b
+
1
)
=
1
\gcd(a-b,ab+1)=1
g
cd
(
a
−
b
,
ab
+
1
)
=
1
and
gcd
(
a
+
b
,
a
b
−
1
)
=
1
\gcd(a+b,ab-1)=1
g
cd
(
a
+
b
,
ab
−
1
)
=
1
. Prove that
(
a
−
b
)
2
+
(
a
b
+
1
)
2
(a-b)^2+(ab+1)^2
(
a
−
b
)
2
+
(
ab
+
1
)
2
is not a perfect square.
4
1
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Iran National Math Olympiad-Day 2-Problem 4
Let
P
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
P(x)=ax^3+bx^2+cx+d
P
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
be a polynomial with real coefficients such that
min
{
d
,
b
+
d
}
>
max
{
∣
c
∣
,
∣
a
+
c
∣
}
\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}
min
{
d
,
b
+
d
}
>
max
{
∣
c
∣
,
∣
a
+
c
∣
}
Prove that
P
(
x
)
P(x)
P
(
x
)
do not have a real root in
[
−
1
,
1
]
[-1,1]
[
−
1
,
1
]
.