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International Contests
Balkan MO Shortlist
2007 Balkan MO Shortlist
2007 Balkan MO Shortlist
Part of
Balkan MO Shortlist
Subcontests
(16)
A4
1
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The sequence contains infinitely many terms of the given form
Show that the sequence \begin{align*} a_n = \left \lfloor \left( \sqrt[3]{n-2} + \sqrt[3]{n+3} \right)^3 \right \rfloor \end{align*} contains infinitely many terms of the form
a
n
a
n
a_n^{a_n}
a
n
a
n
A2
1
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Find all values of a such that the given polynomial has three real roots
Find all values of
a
∈
R
a \in \mathbb{R}
a
∈
R
for which the polynomial \begin{align*} f(x)=x^4-2x^3 + \left(5-6a^2 \right)x^2 + \left(2a^2-4 \right)x + \left(a^2 -2 \right)^2 \end{align*} has exactly three real roots.
A8
1
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Prove that a_1 =0 for a sequence of real numbers
Let
c
>
2
c>2
c
>
2
and
a
0
,
a
1
,
…
a_0,a_1, \ldots
a
0
,
a
1
,
…
be a sequence of real numbers such that \begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*} for any
n
n
n
∈
\in
∈
N
\mathbb{N}
N
. Prove,
a
1
=
0
a_1=0
a
1
=
0
A1
1
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Fin the minimum and maximum of the given function
Find the minimum and maximum value of the function \begin{align*} f(x,y)=ax^2+cy^2 \end{align*} Under the condition
a
x
2
−
b
x
y
+
c
y
2
=
d
ax^2-bxy+cy^2=d
a
x
2
−
b
x
y
+
c
y
2
=
d
, where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
are positive real numbers such that
b
2
−
4
a
c
<
0
b^2 -4ac <0
b
2
−
4
a
c
<
0
N5
1
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sum of q_ibeta_i is greater than p^2
Let
p
≥
5
p \geq 5
p
≥
5
be a prime and let \begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*} where
q
i
q_i
q
i
are primes. Prove, \begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}
N4
1
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Problem on Infinite Arithmetic Progressions
Find all infinite arithmetic progressions formed with positive integers such that there exists a number
N
∈
N
N \in \mathbb{N}
N
∈
N
, such that for any prime
p
p
p
,
p
>
N
p > N
p
>
N
, the
p
p
p
-th term of the progression is also prime.
N1
1
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System of two diophantine equations
Solve the given system in prime numbers \begin{align*} x^2+yu = (x+u)^v \end{align*} \begin{align*} x^2+yz=u^4 \end{align*}
G4
1
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h<=2H , altitudes inequality of an acute triangle inscribed in a triangle
Points
M
,
N
M,N
M
,
N
and
P
P
P
on the sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
of
△
A
B
C
\vartriangle ABC
△
A
BC
are such that
△
M
N
P
\vartriangle MNP
△
MNP
is acute. Denote by
h
h
h
and
H
H
H
the lengths of the shortest altitude of
△
A
B
C
\vartriangle ABC
△
A
BC
and the longest altitude of
△
M
N
P
\vartriangle MNP
△
MNP
. Prove that
h
≤
2
H
h \le 2H
h
≤
2
H
.
G1
1
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perpendicularity wanted related to tangents in a circle
Let
ω
\omega
ω
be a circle with center
O
O
O
and let
A
A
A
be a point outside
ω
\omega
ω
. The tangents from
A
A
A
touch
ω
\omega
ω
at points
B
B
B
, and
C
C
C
. Let
D
D
D
be the point at which the line
A
O
AO
A
O
intersects the circle such that
O
O
O
is between
A
A
A
and
D
D
D
. Denote by
X
X
X
the orthogonal projection of
B
B
B
onto
C
D
CD
C
D
, by
Y
Y
Y
the midpoint of the segment
B
X
BX
BX
and by
Z
Z
Z
the second point of intersection of the line
D
Y
DY
D
Y
with
ω
\omega
ω
. Prove that
Z
A
ZA
Z
A
and
Z
C
ZC
ZC
are perpendicular to each other.
A5
1
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function hard
find all the function
f
,
g
:
R
→
R
f,g:R\rightarrow R
f
,
g
:
R
→
R
such that (1)for every
x
,
y
∈
R
x,y\in R
x
,
y
∈
R
we have
f
(
x
g
(
y
+
1
)
)
+
y
=
x
f
(
y
)
+
f
(
x
+
g
(
y
)
)
f(xg(y+1))+y=xf(y)+f(x+g(y))
f
(
xg
(
y
+
1
))
+
y
=
x
f
(
y
)
+
f
(
x
+
g
(
y
))
(2)
f
(
0
)
+
g
(
0
)
=
0
f(0)+g(0)=0
f
(
0
)
+
g
(
0
)
=
0
N2
1
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Bulgaria TST, IMO 2007- 1-st test, problem 3
Prove that there are no distinct positive integers
x
x
x
and
y
y
y
such that
x
2007
+
y
!
=
y
2007
+
x
!
x^{2007} + y! = y^{2007} + x!
x
2007
+
y
!
=
y
2007
+
x
!
N3
1
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bulgaria tst 2007
i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is: Find all positive integers m for which for all
α
,
β
∈
Z
−
{
0
}
\alpha,\beta \in \mathbb{Z}-\{0\}
α
,
β
∈
Z
−
{
0
}
2
m
α
m
−
(
α
+
β
)
m
−
(
α
−
β
)
m
3
α
2
+
β
2
∈
Z
\frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z}
3
α
2
+
β
2
2
m
α
m
−
(
α
+
β
)
m
−
(
α
−
β
)
m
∈
Z
A3
1
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Fourier coefficients inequality
For
n
∈
N
n\in\mathbb{N}
n
∈
N
,
n
≥
2
n\geq 2
n
≥
2
,
a
i
,
b
i
∈
R
a_{i}, b_{i}\in\mathbb{R}
a
i
,
b
i
∈
R
,
1
≤
i
≤
n
1\leq i\leq n
1
≤
i
≤
n
, such that
∑
i
=
1
n
a
i
2
=
∑
i
=
1
n
b
i
2
=
1
,
∑
i
=
1
n
a
i
b
i
=
0.
\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0.
i
=
1
∑
n
a
i
2
=
i
=
1
∑
n
b
i
2
=
1
,
i
=
1
∑
n
a
i
b
i
=
0.
Prove that
(
∑
i
=
1
n
a
i
)
2
+
(
∑
i
=
1
n
b
i
)
2
≤
n
.
\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n.
(
i
=
1
∑
n
a
i
)
2
+
(
i
=
1
∑
n
b
i
)
2
≤
n
.
Cezar Lupu & Tudorel Lupu
C3
1
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Travel companies, true from 11 up
Three travel companies provide transportation between
n
n
n
cities, such that each connection between a pair of cities is covered by one company only. Prove that, for
n
≥
11
n \geq 11
n
≥
11
, there must exist a round-trip through some four cities, using the services of a same company, while for
n
<
11
n < 11
n
<
11
this is not anymore necessarily true.Dan Schwarz
G3
1
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Convex pentagon and equal areas
Let
A
1
A
2
A
3
A
4
A
5
A_{1}A_{2}A_{3}A_{4}A_{5}
A
1
A
2
A
3
A
4
A
5
be a convex pentagon, such that [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}]. Prove that there exists a point
M
M
M
in the plane of the pentagon such that [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}]. Here
[
X
Y
Z
]
[XYZ]
[
X
Y
Z
]
stands for the area of the triangle
Δ
X
Y
Z
\Delta XYZ
Δ
X
Y
Z
.
C2
1
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Find max of Im(f)
Let
F
\mathcal{F}
F
be the set of all the functions
f
:
P
(
S
)
⟶
R
f : \mathcal{P}(S) \longrightarrow \mathbb{R}
f
:
P
(
S
)
⟶
R
such that for all
X
,
Y
⊆
S
X, Y \subseteq S
X
,
Y
⊆
S
, we have
f
(
X
∩
Y
)
=
min
(
f
(
X
)
,
f
(
Y
)
)
f(X \cap Y) = \min (f(X), f(Y))
f
(
X
∩
Y
)
=
min
(
f
(
X
)
,
f
(
Y
))
, where
S
S
S
is a finite set (and
P
(
S
)
\mathcal{P}(S)
P
(
S
)
is the set of its subsets). Find
max
f
∈
F
∣
Im
(
f
)
∣
.
\max_{f \in \mathcal{F}}| \textrm{Im}(f) |.
f
∈
F
max
∣
Im
(
f
)
∣.