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Contests
National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2023 Iran MO (2nd Round)
2023 Iran MO (2nd Round)
Part of
Iran MO (2nd Round)
Subcontests
(6)
P5
1
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2023 Iran MO 2nd round P5
5. We call
(
P
n
)
n
∈
N
(P_n)_{n\in \mathbb{N}}
(
P
n
)
n
∈
N
an arithmetic sequence with common difference
Q
(
x
)
Q(x)
Q
(
x
)
if
∀
n
:
P
n
+
1
=
P
n
+
Q
\forall n: P_{n+1} = P_n + Q
∀
n
:
P
n
+
1
=
P
n
+
Q
\newline
We have an arithmetic sequence with a common difference
Q
(
x
)
Q(x)
Q
(
x
)
and the first term
P
(
x
)
P(x)
P
(
x
)
such that
P
,
Q
P,Q
P
,
Q
are monic polynomials with integer coefficients and don't share an integer root. Each term of the sequence has at least one integer root. Prove that:
\newline
a)
P
(
x
)
P(x)
P
(
x
)
is divisible by
Q
(
x
)
Q(x)
Q
(
x
)
\newline
b)
deg
(
P
(
x
)
Q
(
x
)
)
=
1
\text{deg}(\frac{P(x)}{Q(x)}) = 1
deg
(
Q
(
x
)
P
(
x
)
)
=
1
P6
1
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2023 Iran MO 2nd round P6
6. Circles
W
1
W_{1}
W
1
and
W
2
W_{2}
W
2
with equal radii are given. Let
P
P
P
,
Q
Q
Q
be the intersection of the circles. points
B
B
B
and
C
C
C
are on
W
1
W_{1}
W
1
and
W
2
W_{2}
W
2
such that they are inside
W
2
W_{2}
W
2
and
W
1
W_{1}
W
1
respectively. Points
X
X
X
,
Y
Y
Y
≠
\neq
=
P
P
P
are on
W
1
W_{1}
W
1
and
W
2
W_{2}
W
2
respectively, such that
∠
B
P
Q
=
∠
B
Y
Q
\angle{BPQ}=\angle{BYQ}
∠
BPQ
=
∠
B
Y
Q
and
∠
C
P
Q
=
∠
C
X
Q
\angle{CPQ}=\angle{CXQ}
∠
CPQ
=
∠
CXQ
.Denote by
S
S
S
as the other intersection of
(
Y
P
B
)
(YPB)
(
Y
PB
)
and
(
X
P
C
)
(XPC)
(
XPC
)
. Prove that
Q
S
,
B
C
,
X
Y
QS,BC,XY
QS
,
BC
,
X
Y
are concurrent.
P3
1
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2023 Iran MO 2nd round P3
3. We have a
n
×
n
n \times n
n
×
n
board. We color the unit square
(
i
,
j
)
(i,j)
(
i
,
j
)
black if
i
=
j
i=j
i
=
j
, red if
i
<
j
i<j
i
<
j
and green if
i
>
j
i>j
i
>
j
. Let
a
i
,
j
a_{i,j}
a
i
,
j
be the color of the unit square
(
i
,
j
)
(i,j)
(
i
,
j
)
. In each move we switch two rows and write down the
n
n
n
-tuple
(
a
1
,
1
,
a
2
,
2
,
⋯
,
a
n
,
n
)
(a_{1,1},a_{2,2},\cdots,a_{n,n})
(
a
1
,
1
,
a
2
,
2
,
⋯
,
a
n
,
n
)
. How many
n
n
n
-tuples can we obtain by repeating this process? (note that the order of the numbers are important)
P4
1
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2023 Iran MO 2nd round P4
4. A positive integer n is given.Find the smallest
k
k
k
such that we can fill a
3
∗
k
3*k
3
∗
k
gird with non-negative integers such that:
\newline
i
i
i
) Sum of the numbers in each column is
n
n
n
.
i
i
ii
ii
) Each of the numbers
0
,
1
,
…
,
n
0,1,\dots,n
0
,
1
,
…
,
n
appears at least once in each row.
P2
1
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2023 Iran MO 2nd round P2
2. Prove that for any
2
≤
n
∈
N
2\le n \in \mathbb{N}
2
≤
n
∈
N
there exists positive integers
a
1
,
a
2
,
⋯
,
a
n
a_1,a_2,\cdots,a_n
a
1
,
a
2
,
⋯
,
a
n
such that
∀
i
≠
j
:
gcd
(
a
i
,
a
j
)
=
1
\forall i\neq j: \text{gcd}(a_i,a_j) = 1
∀
i
=
j
:
gcd
(
a
i
,
a
j
)
=
1
and
∀
i
:
a
i
≥
1402
\forall i: a_i \ge 1402
∀
i
:
a
i
≥
1402
and the given relation holds.
[
a
1
a
2
]
+
[
a
2
a
3
]
+
⋯
+
[
a
n
a
1
]
=
[
a
2
a
1
]
+
[
a
3
a
2
]
+
⋯
+
[
a
1
a
n
]
[\frac{a_1}{a_2}]+[\frac{a_2}{a_3}]+\cdots+[\frac{a_n}{a_1}] = [\frac{a_2}{a_1}]+[\frac{a_3}{a_2}]+\cdots+[\frac{a_1}{a_n}]
[
a
2
a
1
]
+
[
a
3
a
2
]
+
⋯
+
[
a
1
a
n
]
=
[
a
1
a
2
]
+
[
a
2
a
3
]
+
⋯
+
[
a
n
a
1
]
P1
1
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2023 Iran MO 2nd round P1
1. In right triangle
A
B
C
ABC
A
BC
with \angle{A}= \textdegree{90}, point
P
P
P
is chosen.
D
∈
B
C
D \in BC
D
∈
BC
such that
P
D
⊥
B
C
PD \perp BC
P
D
⊥
BC
. Let the intersection of
P
D
PD
P
D
with
A
B
AB
A
B
and
A
C
AC
A
C
be
E
E
E
and
F
F
F
respectively. Denote by
X
X
X
and
Y
Y
Y
as the intersection of
(
A
P
E
)
(APE)
(
A
PE
)
and
(
A
P
F
)
(APF)
(
A
PF
)
with
B
P
BP
BP
and
C
P
CP
CP
respectively. Prove that
C
X
,
B
Y
,
P
D
CX,BY,PD
CX
,
B
Y
,
P
D
are concurrent.