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Contests
National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2024 Iran MO (3rd Round)
2024 Iran MO (3rd Round)
Part of
Iran MO (3rd Round)
Subcontests
(6)
5
1
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Tangent circles in a parallelogram configuration
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and let
A
X
AX
A
X
and
A
Y
AY
A
Y
be the altitudes from
A
A
A
to
C
B
,
C
D
CB, CD
CB
,
C
D
, respectively. A line
ℓ
⊥
X
Y
\ell \perp XY
ℓ
⊥
X
Y
bisects
A
X
AX
A
X
and meets
A
B
,
B
C
AB, BC
A
B
,
BC
at
K
,
L
K, L
K
,
L
. Similarly, a line
d
⊥
X
Y
d \perp XY
d
⊥
X
Y
bisects
A
Y
AY
A
Y
and meets
D
A
,
D
C
DA, DC
D
A
,
D
C
at
P
,
Q
P, Q
P
,
Q
. Show that the circumcircles of
△
B
K
L
\triangle BKL
△
B
K
L
and
△
D
P
Q
\triangle DPQ
△
D
PQ
are tangent to each other.
2
5
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4
1
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Cute division problem
For a given positive integer number
n
n
n
find all subsets
{
r
0
,
r
1
,
⋯
,
r
n
}
⊂
N
\{r_0,r_1,\cdots,r_n\}\subset \mathbb{N}
{
r
0
,
r
1
,
⋯
,
r
n
}
⊂
N
such that
n
n
+
n
n
−
1
+
⋯
+
1
∣
n
r
n
+
⋯
+
n
r
0
.
n^n+n^{n-1}+\cdots+1 | n^{r_n}+\cdots+ n^{r_0}.
n
n
+
n
n
−
1
+
⋯
+
1∣
n
r
n
+
⋯
+
n
r
0
.
6
1
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Polynomial and sequence
Sequence of positive integers
{
x
k
}
k
≥
1
\{x_k\}_{k\geq 1}
{
x
k
}
k
≥
1
is given such that
x
1
=
1
x_1=1
x
1
=
1
and for all
n
≥
1
n\geq 1
n
≥
1
we have
x
n
+
1
2
+
P
(
n
)
=
x
n
x
n
+
2
x_{n+1}^2+P(n)=x_n x_{n+2}
x
n
+
1
2
+
P
(
n
)
=
x
n
x
n
+
2
where
P
(
x
)
P(x)
P
(
x
)
is a polynomial with non-negative integer coefficients. Prove that
P
(
x
)
P(x)
P
(
x
)
is the constant polynomial.
3
5
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1
5
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