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IFYM Int. Fest. of Young Mathematicians, Sozopol
2013 IFYM, Sozopol
2013 IFYM, Sozopol
Part of
IFYM Int. Fest. of Young Mathematicians, Sozopol
Subcontests
(8)
7
4
Show problems
1
4
Show problems
5
3
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Problem 5 of Second round
Determine all increasing sequences
{
a
n
}
n
=
1
∞
\{a_n\}_{n=1}^\infty
{
a
n
}
n
=
1
∞
of natural numbers with the following property: for each two natural numbers
i
i
i
and
j
j
j
(not necessarily different), the numbers
i
+
j
i+j
i
+
j
and
a
i
+
a
j
a_i+a_j
a
i
+
a
j
have an equal number of distinct natural divisors.
Polynomial equation
Find all polynomilals
P
P
P
with real coefficients, such that
(
x
+
1
)
P
(
x
−
1
)
+
(
x
−
1
)
P
(
x
+
1
)
=
2
x
P
(
x
)
(x+1)P(x-1)+(x-1)P(x+1)=2xP(x)
(
x
+
1
)
P
(
x
−
1
)
+
(
x
−
1
)
P
(
x
+
1
)
=
2
x
P
(
x
)
2n+7 | n! -1
Find all positive integers
n
n
n
satisfying
2
n
+
7
∣
n
!
−
1
2n+7 \mid n! -1
2
n
+
7
∣
n
!
−
1
.
2
4
Show problems
4
4
Show problems
3
4
Show problems
6
4
Show problems
8
3
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Problem 8 of Second round
Let
K
K
K
be a point on the angle bisector, such that
∠
B
K
L
=
∠
K
B
L
=
3
0
∘
\angle BKL=\angle KBL=30^\circ
∠
B
K
L
=
∠
K
B
L
=
3
0
∘
. The lines
A
B
AB
A
B
and
C
K
CK
C
K
intersect in point
M
M
M
and lines
A
C
AC
A
C
and
B
K
BK
B
K
intersect in point
N
N
N
. Determine
∠
A
M
N
\angle AMN
∠
A
MN
.
Problem 8 of Third round
The irrational numbers
α
,
β
,
γ
,
δ
\alpha ,\beta ,\gamma ,\delta
α
,
β
,
γ
,
δ
are such that
∀
\forall
∀
n
∈
N
n\in \mathbb{N}
n
∈
N
:
[
n
α
]
.
[
n
β
]
=
[
n
γ
]
.
[
n
δ
]
[n\alpha ].[n\beta ]=[n\gamma ].[n\delta ]
[
n
α
]
.
[
n
β
]
=
[
nγ
]
.
[
n
δ
]
. Is it true that the sets
{
α
,
β
}
\{ \alpha ,\beta \}
{
α
,
β
}
and
{
γ
,
δ
}
\{ \gamma ,\delta \}
{
γ
,
δ
}
are equal?
Korea Third Round (FKMO) 2012 #1
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be positive real numbers. Prove that
2
x
2
+
x
y
(
y
+
z
x
+
z
)
2
+
2
y
2
+
y
z
(
z
+
x
y
+
x
)
2
+
2
z
2
+
z
x
(
x
+
y
z
+
y
)
2
≥
1
\frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1
(
y
+
z
x
+
z
)
2
2
x
2
+
x
y
+
(
z
+
x
y
+
x
)
2
2
y
2
+
yz
+
(
x
+
yz
+
y
)
2
2
z
2
+
z
x
≥
1