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Problems
Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2002 Iran MO (2nd round)
2002 Iran MO (2nd round)
Part of
Iran MO (2nd Round)
Subcontests
(5)
6
1
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Prove that the graph G is regular
Let
G
G
G
be a simple graph with
100
100
100
edges on
20
20
20
vertices. Suppose that we can choose a pair of disjoint edges in
4050
4050
4050
ways. Prove that
G
G
G
is regular.
5
1
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Multiple root of a polynomial
Let
δ
\delta
δ
be a symbol such that
δ
≠
0
\delta \neq 0
δ
=
0
and
δ
2
=
0
\delta^2 = 0
δ
2
=
0
. Define
R
[
δ
]
=
{
a
+
b
δ
∣
a
,
b
∈
R
}
\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}
R
[
δ
]
=
{
a
+
b
δ
∣
a
,
b
∈
R
}
, where
a
+
b
δ
=
c
+
d
δ
a+ b \delta = c+ d \delta
a
+
b
δ
=
c
+
d
δ
if and only if
a
=
c
a = c
a
=
c
and
b
=
d
b = d
b
=
d
, and define
(
a
+
b
δ
)
+
(
c
+
d
δ
)
=
(
a
+
c
)
+
(
b
+
d
)
δ
,
(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,
(
a
+
b
δ
)
+
(
c
+
d
δ
)
=
(
a
+
c
)
+
(
b
+
d
)
δ
,
(
a
+
b
δ
)
⋅
(
c
+
d
δ
)
=
a
c
+
(
a
d
+
b
c
)
δ
.
(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.
(
a
+
b
δ
)
⋅
(
c
+
d
δ
)
=
a
c
+
(
a
d
+
b
c
)
δ
.
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with real coefficients. Show that
P
(
x
)
P(x)
P
(
x
)
has a multiple real root if and only if
P
(
x
)
P(x)
P
(
x
)
has a non-real root in
R
[
δ
]
.
\mathbb R[\delta].
R
[
δ
]
.
4
1
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Prove that there is a fixed point P
Let
A
A
A
and
B
B
B
be two fixed points in the plane. Consider all possible convex quadrilaterals
A
B
C
D
ABCD
A
BC
D
with
A
B
=
B
C
,
A
D
=
D
C
AB = BC, AD = DC
A
B
=
BC
,
A
D
=
D
C
, and
∠
A
D
C
=
9
0
∘
\angle ADC = 90^\circ
∠
A
D
C
=
9
0
∘
. Prove that there is a fixed point
P
P
P
such that, for every such quadrilateral
A
B
C
D
ABCD
A
BC
D
on the same side of
A
B
AB
A
B
, the line
D
C
DC
D
C
passes through
P
.
P.
P
.
2
1
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Rectangle is partitioned into finitely many small rectangles
A rectangle is partitioned into finitely many small rectangles. We call a point a cross point if it belongs to four different small rectangles. We call a segment on the obtained diagram maximal if there is no other segment containing it. Show that the number of maximal segments plus the number of cross points is
3
3
3
more than the number of small rectangles.
3
1
Hide problems
nice problem(due hard)
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with
∠
A
B
C
=
∠
A
D
C
=
13
5
∘
\angle ABC = \angle ADC = 135^\circ
∠
A
BC
=
∠
A
D
C
=
13
5
∘
, points
M
M
M
and
N
N
N
are taken on the rays
A
B
AB
A
B
and
A
D
AD
A
D
respectively such that
∠
M
C
D
=
∠
N
C
B
=
9
0
∘
\angle MCD = \angle NCB = 90^\circ
∠
MC
D
=
∠
NCB
=
9
0
∘
. The circumcircles of triangles
A
M
N
AMN
A
MN
and
A
B
D
ABD
A
B
D
intersect at
A
A
A
and
K
K
K
. Prove that
A
K
⊥
K
C
.
AK \perp KC.
A
K
⊥
K
C
.