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Contests
International Contests
Balkan MO Shortlist
2009 Balkan MO Shortlist
2009 Balkan MO Shortlist
Part of
Balkan MO Shortlist
Subcontests
(15)
A2
1
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Prove the existence of the configuration and ...
Let
A
B
C
D
ABCD
A
BC
D
be a square and points
M
M
M
∈
\in
∈
B
C
BC
BC
,
N
∈
C
D
N \in CD
N
∈
C
D
,
P
P
P
∈
\in
∈
D
A
DA
D
A
, such that
∠
B
A
M
\angle BAM
∠
B
A
M
=
=
=
x
x
x
,
∠
C
M
N
\angle CMN
∠
CMN
=
=
=
2
x
2x
2
x
,
∠
D
N
P
\angle DNP
∠
D
NP
=
=
=
3
x
3x
3
x
[*] Show that, for any
x
∈
(
0
,
π
8
)
x \in (0, \tfrac{\pi}{8} )
x
∈
(
0
,
8
π
)
, such a configuration exists [*] Determine the number of angles
x
∈
(
0
,
π
8
)
x \in ( 0, \tfrac{\pi}{8} )
x
∈
(
0
,
8
π
)
for which
∠
A
P
B
=
4
x
\angle APB =4x
∠
A
PB
=
4
x
A8
1
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Prove the given inequality and how that K is non-negative
For every positive integer
m
m
m
and for all non-negative real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
denote \begin{align*} K_m =x(x-y)^m (x-z)^m + y (y-x)^m (y-z)^m + z(z-x)^m (z-y)^m \end{align*}[*] Prove that
K
m
≥
0
K_m \geq 0
K
m
≥
0
for every odd positive integer
m
m
m
[*] Let
M
M
M
=
∏
c
y
c
(
x
−
y
)
2
= \prod_{cyc} (x-y)^2
=
∏
cyc
(
x
−
y
)
2
. Prove,
K
7
+
M
2
K
1
≥
M
K
4
K_7+M^2 K_1 \geq M K_4
K
7
+
M
2
K
1
≥
M
K
4
A1
1
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Find all possible values of
Let
N
∈
N
N \in \mathbb{N}
N
∈
N
and
x
k
∈
[
−
1
,
1
]
x_k \in [-1,1]
x
k
∈
[
−
1
,
1
]
,
1
≤
k
≤
N
1 \le k \le N
1
≤
k
≤
N
such that
∑
k
=
1
N
x
k
=
s
\sum_{k=1}^N x_k =s
∑
k
=
1
N
x
k
=
s
. Find all possible values of
∑
k
=
1
N
∣
x
k
∣
\sum_{k=1}^N |x_k|
∑
k
=
1
N
∣
x
k
∣
A7
1
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Prove that the given polynomial is irreducible
Let
n
≥
2
n\geq 2
n
≥
2
be a positive integer and \begin{align*} P(x) = c_0 X^n + c_1 X^{n-1} + \ldots + c_{n-1} X +c_n \end{align*} be a polynomial with integer coefficients, such that
∣
c
n
∣
\mid c_n \mid
∣
c
n
∣
is a prime number and \begin{align*} |c_0| + |c_1| + \ldots + |c_{n-1}| < |c_n| \end{align*} Prove that the polynomial
P
(
X
)
P(X)
P
(
X
)
is irreducible in the
Z
[
x
]
\mathbb{Z}[x]
Z
[
x
]
A5
1
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An inequality on the coefficients of a monic polynomial
Given the monic polynomial \begin{align*} P(x) = x^N +a_{N-1}x^{N-1} + \ldots + a_1 x + a_0 \in \mathbb{R}[x] \end{align*} of even degree
N
N
N
=
=
=
2
n
2n
2
n
and having all real positive roots
x
i
x_i
x
i
, for
1
≤
i
≤
N
1 \le i \le N
1
≤
i
≤
N
. Prove, for any
c
c
c
∈
\in
∈
[
0
,
min
1
≤
i
≤
N
{
x
i
}
)
[0, \underset{1 \le i \le N}{\min} \{x_i \} )
[
0
,
1
≤
i
≤
N
min
{
x
i
})
, the following inequality \begin{align*} c + \sqrt[N]{P(c)} \le \sqrt[N]{a_0} \end{align*}
A3
1
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Prove the upper bound on the sequence x_n
Denote by
S
(
x
)
S(x)
S
(
x
)
the sum of digits of positive integer
x
x
x
written in decimal notation. For
k
k
k
a fixed positive integer, define a sequence
(
x
n
)
n
≥
1
(x_n)_{n \geq 1}
(
x
n
)
n
≥
1
by
x
1
=
1
x_1=1
x
1
=
1
and
x
n
+
1
x_{n+1}
x
n
+
1
=
=
=
S
(
k
x
n
)
S(kx_n)
S
(
k
x
n
)
for all positive integers
n
n
n
. Prove that
x
n
x_n
x
n
<
<
<
27
k
27 \sqrt{k}
27
k
for all positive integer
n
n
n
.
N3
1
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product of all integers of form i^3+1 is a perfect square
Determine all integers
1
≤
m
,
1
≤
n
≤
2009
1 \le m, 1 \le n \le 2009
1
≤
m
,
1
≤
n
≤
2009
, for which \begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}
N1
1
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Diophantine equation of degree 6
Solve the given equation in integers \begin{align*} y^3=8x^6+2x^3y-y^2 \end{align*}
C2
1
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On the Subsets of the sets 1,2,...,n
Let
A
1
,
A
2
,
…
,
A
m
A_1, A_2, \ldots , A_m
A
1
,
A
2
,
…
,
A
m
be subsets of the set
{
1
,
2
,
…
,
n
}
\{ 1,2, \ldots , n \}
{
1
,
2
,
…
,
n
}
, such that the cardinal of each subset
A
i
A_i
A
i
, such
1
≤
i
≤
m
1 \le i \le m
1
≤
i
≤
m
is not divisible by
30
30
30
, while the cardinal of each of the subsets
A
i
∩
A
j
A_i \cap A_j
A
i
∩
A
j
for
1
≤
i
,
j
≤
m
1 \le i,j \le m
1
≤
i
,
j
≤
m
,
i
≠
j
i \neq j
i
=
j
is divisible by
30
30
30
. Prove \begin{align*} 2m - \left \lfloor \frac{m}{30} \right \rfloor \le 3n \end{align*}
G5
1
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(AECZ) = (EBZD) = (ABCD) , equal areas
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral and
S
S
S
an arbitrary point in its interior. Let also
E
E
E
be the symmetric point of
S
S
S
with respect to the midpoint
K
K
K
of the side
A
B
AB
A
B
and let
Z
Z
Z
be the symmetric point of
S
S
S
with respect to the midpoint
L
L
L
of the side
C
D
CD
C
D
. Prove that
(
A
E
C
Z
)
=
(
E
B
Z
D
)
=
(
A
B
C
D
)
(AECZ) = (EBZD) = (ABCD)
(
A
ECZ
)
=
(
EBZ
D
)
=
(
A
BC
D
)
.
G3
1
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circumcenter of projections of P on sides of convex ABCD lies on MN, midpoints
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, and
P
P
P
be a point in its interior. The projections of
P
P
P
on the sides of the quadrilateral lie on a circle with center
O
O
O
. Show that
O
O
O
lies on the line through the midpoints of
A
C
AC
A
C
and
B
D
BD
B
D
.
G1
1
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PL^2 = DQ x PQ wanted, DQ = DK, <ADB = 45^o, <KDP = 30^o given
In the triangle
A
B
C
,
∠
B
A
C
ABC, \angle BAC
A
BC
,
∠
B
A
C
is acute, the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
meets
B
C
BC
BC
at
D
,
K
D, K
D
,
K
is the foot of the perpendicular from
B
B
B
to
A
C
AC
A
C
, and
∠
A
D
B
=
4
5
o
\angle ADB = 45^o
∠
A
D
B
=
4
5
o
. Point
P
P
P
lies between
K
K
K
and
C
C
C
such that
∠
K
D
P
=
3
0
o
\angle KDP = 30^o
∠
KD
P
=
3
0
o
. Point
Q
Q
Q
lies on the ray
D
P
DP
D
P
such that
D
Q
=
D
K
DQ = DK
D
Q
=
DK
. The perpendicular at
P
P
P
to
A
C
AC
A
C
meets
K
D
KD
KD
at
L
L
L
. Prove that
P
L
2
=
D
Q
⋅
P
Q
PL^2 = DQ \cdot PQ
P
L
2
=
D
Q
⋅
PQ
.
A6
1
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Functional equation with a) and b)
We denote the set of nonzero integers and the set of non-negative integers with
Z
∗
\mathbb Z^*
Z
∗
and
N
0
\mathbb N_0
N
0
, respectively. Find all functions
f
:
Z
∗
→
N
0
f:\mathbb Z^* \to \mathbb N_0
f
:
Z
∗
→
N
0
such that:
a
)
a)
a
)
f
(
a
+
b
)
≥
m
i
n
(
f
(
a
)
,
f
(
b
)
)
f(a+b)\geq min(f(a), f(b))
f
(
a
+
b
)
≥
min
(
f
(
a
)
,
f
(
b
))
for all
a
,
b
a,b
a
,
b
in
Z
∗
\mathbb Z^*
Z
∗
for which
a
+
b
a+b
a
+
b
is in
Z
∗
\mathbb Z^*
Z
∗
.
b
)
b)
b
)
f
(
a
b
)
=
f
(
a
)
+
f
(
b
)
f(ab)=f(a)+f(b)
f
(
ab
)
=
f
(
a
)
+
f
(
b
)
for all
a
,
b
a,b
a
,
b
in
Z
∗
\mathbb Z^*
Z
∗
.
G6
1
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Circumradii of MEF and NEF are of same length
Two circles
O
1
O_1
O
1
and
O
2
O_2
O
2
intersect each other at
M
M
M
and
N
N
N
. The common tangent to two circles nearer to
M
M
M
touch
O
1
O_1
O
1
and
O
2
O_2
O
2
at
A
A
A
and
B
B
B
respectively. Let
C
C
C
and
D
D
D
be the reflection of
A
A
A
and
B
B
B
respectively with respect to
M
M
M
. The circumcircle of the triangle
D
C
M
DCM
D
CM
intersect circles
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively at points
E
E
E
and
F
F
F
(both distinct from
M
M
M
). Show that the circumcircles of triangles
M
E
F
MEF
MEF
and
N
E
F
NEF
NEF
have same radius length.
G2
1
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Cyclic hexagon
If
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is a convex cyclic hexagon, then its diagonals
A
D
AD
A
D
,
B
E
BE
BE
,
C
F
CF
CF
are concurrent if and only if
A
B
B
C
⋅
C
D
D
E
⋅
E
F
F
A
=
1
\frac{AB}{BC}\cdot \frac{CD}{DE}\cdot \frac{EF}{FA}=1
BC
A
B
⋅
D
E
C
D
⋅
F
A
EF
=
1
. Alternative version. Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a hexagon inscribed in a circle. Then, the lines
A
D
AD
A
D
,
B
E
BE
BE
,
C
F
CF
CF
are concurrent if and only if
A
B
⋅
C
D
⋅
E
F
=
B
C
⋅
D
E
⋅
F
A
AB\cdot CD\cdot EF=BC\cdot DE\cdot FA
A
B
⋅
C
D
⋅
EF
=
BC
⋅
D
E
⋅
F
A
.