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Contests
International Contests
Balkan MO Shortlist
2013 Balkan MO Shortlist
2013 Balkan MO Shortlist
Part of
Balkan MO Shortlist
Subcontests
(22)
N9
1
Hide problems
x_1^{x_2}=x_2^{x_3}=...=x_{n-1}^{x_n}=x_n^{x_1} solve in Q+
Let
n
≥
2
n\ge 2
n
≥
2
be a given integer. Determine all sequences
x
1
,
.
.
.
,
x
n
x_1,...,x_n
x
1
,
...
,
x
n
of positive rational numbers such that
x
1
x
2
=
x
2
x
3
=
.
.
.
=
x
n
−
1
x
n
=
x
n
x
1
x_1^{x_2}=x_2^{x_3}=...=x_{n-1}^{x_n}=x_n^{x_1}
x
1
x
2
=
x
2
x
3
=
...
=
x
n
−
1
x
n
=
x
n
x
1
N8
1
Hide problems
a+b+c+d=0 and ac+bd=0, iff a-b divides 2ab
Suppose that
a
a
a
and
b
b
b
are integers. Prove that there are integers
c
c
c
and
d
d
d
such that
a
+
b
+
c
+
d
=
0
a+b+c+d=0
a
+
b
+
c
+
d
=
0
and
a
c
+
b
d
=
0
ac+bd=0
a
c
+
b
d
=
0
, if and only if
a
−
b
a-b
a
−
b
divides
2
a
b
2ab
2
ab
.
N7
1
Hide problems
set of elements where for each 2 elements their gcd equals difference
Two distinct positive integers are called close if their greatest common divisor equals their difference. Show that for any
n
n
n
, there exists a set
S
S
S
of
n
n
n
elements such that any two elements of
S
S
S
are close.
N6
1
Hide problems
p^{q-1}- q^{p-1}=4n^3 diophantine without solution for primes p \ne q
Prove that there do not exist distinct prime numbers
p
p
p
and
q
q
q
and a positive integer
n
n
n
satisfying the equation
p
q
−
1
−
q
p
−
1
=
4
n
3
p^{q-1}- q^{p-1}=4n^3
p
q
−
1
−
q
p
−
1
=
4
n
3
N5
1
Hide problems
p^{q-1}- q^{p-1}=4n^2 diophantine without solution for primes p \ne q
Prove that there do not exist distinct prime numbers
p
p
p
and
q
q
q
and a positive integer
n
n
n
satisfying the equation
p
q
−
1
−
q
p
−
1
=
4
n
2
p^{q-1}- q^{p-1}=4n^2
p
q
−
1
−
q
p
−
1
=
4
n
2
N4
1
Hide problems
1^{p+2} + 2^{p+2} + ...+(p-1)^{p+2} is divisible by p^2
Let
p
p
p
be a prime number greater than
3
3
3
. Prove that the sum
1
p
+
2
+
2
p
+
2
+
.
.
.
+
(
p
−
1
)
p
+
2
1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}
1
p
+
2
+
2
p
+
2
+
...
+
(
p
−
1
)
p
+
2
is divisible by
p
2
p^2
p
2
.
N3
1
Hide problems
12^x + 13^y - 14^z = 2013^t diophantine in positive integers
Determine all quadruplets (
x
,
y
,
z
,
t
x, y, z, t
x
,
y
,
z
,
t
) of positive integers, such that
1
2
x
+
1
3
y
−
1
4
z
=
201
3
t
12^x + 13^y - 14^z = 2013^t
1
2
x
+
1
3
y
−
1
4
z
=
201
3
t
.
N1
1
Hide problems
find a,b,c>0 when 1/a+1/b+1/c=1/p and a + b + c < 2p\sqrt{p}
Let
p
p
p
be a prime number. Determine all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive integers such that
a
+
b
+
c
<
2
p
p
a + b + c < 2p\sqrt{p}
a
+
b
+
c
<
2
p
p
and
1
a
+
1
b
+
1
c
=
1
p
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{p}
a
1
+
b
1
+
c
1
=
p
1
A7
1
Hide problems
|f(z) - z| <= k, for every k, each k-jumpy map is composition of 1-jumpy maps?
Suppose that
k
k
k
is a positive integer. A bijective map
f
:
Z
→
Z
f : Z \to Z
f
:
Z
→
Z
is said to be
k
k
k
-jumpy if
∣
f
(
z
)
−
z
∣
≤
k
|f(z) - z| \le k
∣
f
(
z
)
−
z
∣
≤
k
for all integers
z
z
z
. Is it that case that for every
k
k
k
, each
k
k
k
-jumpy map is a composition of
1
1
1
-jumpy maps? It is well known that this is the case when the support of the map is finite.
A5
1
Hide problems
x^{2n}+y^{2n}+z^{2n}-xy-yz-zx divides (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}
Determine all positive integers
n
n
n
such that
f
n
(
x
,
y
,
z
)
=
x
2
n
+
y
2
n
+
z
2
n
−
x
y
−
y
z
−
z
x
f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx
f
n
(
x
,
y
,
z
)
=
x
2
n
+
y
2
n
+
z
2
n
−
x
y
−
yz
−
z
x
divides
g
n
(
x
,
y
,
z
)
=
(
x
−
y
)
5
n
+
(
y
−
z
)
5
n
+
(
z
−
x
)
5
n
g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}
g
n
(
x
,
y
,
z
)
=
(
x
−
y
)
5
n
+
(
y
−
z
)
5
n
+
(
z
−
x
)
5
n
, as polynomials in
x
,
y
,
z
x, y, z
x
,
y
,
z
with integer coefficients.
A4
1
Hide problems
1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x), integer pol.
Find all positive integers
n
n
n
such that there exist non-constant polynomials with integer coefficients
f
1
(
x
)
,
.
.
.
,
f
n
(
x
)
f_1(x),...,f_n(x)
f
1
(
x
)
,
...
,
f
n
(
x
)
(not necessarily distinct) and
g
(
x
)
g(x)
g
(
x
)
such that
1
+
∏
k
=
1
n
(
f
k
2
(
x
)
−
1
)
=
(
x
2
+
2013
)
2
g
2
(
x
)
1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)
1
+
k
=
1
∏
n
(
f
k
2
(
x
)
−
1
)
=
(
x
2
+
2013
)
2
g
2
(
x
)
A3
1
Hide problems
(x^2-8x+25)(x^2-16x+100)...(x^2-8nx+25n^2)-1 not product of 2 integer polyn.
Prove that the polynomial
P
(
x
)
=
(
x
2
−
8
x
+
25
)
(
x
2
−
16
x
+
100
)
.
.
.
(
x
2
−
8
n
x
+
25
n
2
)
−
1
P (x) = (x^2- 8x + 25) (x^2 - 16x + 100) ... (x^2 - 8nx + 25n^2)- 1
P
(
x
)
=
(
x
2
−
8
x
+
25
)
(
x
2
−
16
x
+
100
)
...
(
x
2
−
8
n
x
+
25
n
2
)
−
1
,
n
∈
N
∗
n \in N^*
n
∈
N
∗
, cannot be written as the product of two polynomials with integer coefficients of degree greater or equal to
1
1
1
.
A2
1
Hide problems
(16ac +a/c^2b+16c/a^2d+4/ac)(bd +b/256d^2c+d/b^2a+1/64bd) >=81/4
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
are positive real numbers so that
a
b
c
d
=
1
4
abcd = \frac14
ab
c
d
=
4
1
. Prove that holds
(
16
a
c
+
a
c
2
b
+
16
c
a
2
d
+
4
a
c
)
(
b
d
+
b
256
d
2
c
+
d
b
2
a
+
1
64
b
d
)
≥
81
4
\left( 16ac +\frac{a}{c^2b}+\frac{16c}{a^2d}+\frac{4}{ac}\right)\left( bd +\frac{b}{256d^2c}+\frac{d}{b^2a}+\frac{1}{64bd}\right) \ge \frac{81}{4}
(
16
a
c
+
c
2
b
a
+
a
2
d
16
c
+
a
c
4
)
(
b
d
+
256
d
2
c
b
+
b
2
a
d
+
64
b
d
1
)
≥
4
81
When does the equality hold?
A1
1
Hide problems
1/(4+(a+b)^2)+1/(4+(b+c)^2)+1/(4+(c+a)^2)<= 3/8 for ab + bc+ ca = 3
Positive real numbers
a
,
b
,
c
a, b,c
a
,
b
,
c
satisfy
a
b
+
b
c
+
c
a
=
3
ab + bc+ ca = 3
ab
+
b
c
+
c
a
=
3
. Prove the inequality
1
4
+
(
a
+
b
)
2
+
1
4
+
(
b
+
c
)
2
+
1
4
+
(
c
+
a
)
2
≤
3
8
\frac{1}{4+(a+b)^2}+\frac{1}{4+(b+c)^2}+\frac{1}{4+(c+a)^2}\le \frac{3}{8}
4
+
(
a
+
b
)
2
1
+
4
+
(
b
+
c
)
2
1
+
4
+
(
c
+
a
)
2
1
≤
8
3
C5
1
Hide problems
cells of an n x n chessboard are coloured in several colours
The cells of an
n
×
n
n \times n
n
×
n
chessboard are coloured in several colours so that no
2
×
2
2\times 2
2
×
2
square contains four cells of the same colour. A proper path of length
m
m
m
is a sequence
a
1
,
a
2
,
.
.
.
,
a
m
a_1,a_2,..., a_m
a
1
,
a
2
,
...
,
a
m
of distinct cells in which the cells
a
i
a_i
a
i
and
a
i
+
1
a_{i+1}
a
i
+
1
have a common side and are coloured in different colours for all
1
≤
i
<
m
1 \le i < m
1
≤
i
<
m
. Show that there exists a proper path of length
n
n
n
.
C4
1
Hide problems
closed, non-self-intersecting broken line drawn on chseeboard, red squares
A closed, non-self-intersecting broken line
L
L
L
is drawn over a
(
2
n
+
1
)
×
(
2
n
+
1
)
(2n+1) \times (2n+1)
(
2
n
+
1
)
×
(
2
n
+
1
)
chessboard in such a way that the set of L's vertices coincides with the set of the vertices of the board’s squares and every edge in
L
L
L
is a side of some board square. All board squares lying in the interior of
L
L
L
are coloured in red. Prove that the number of neighbouring pairs of red squares in every row of the board is even.
C3
1
Hide problems
spider moves along a square grid
The square
A
B
C
D
ABCD
A
BC
D
is divided into
n
2
n^2
n
2
equal small (elementary) squares by parallel lines to its sides, (see the figure for the case
n
=
4
n = 4
n
=
4
). A spider starts from point
A
A
A
and moving only to the right and up tries to arrive at point
C
C
C
. Every ” movement” of the spider consists of: ”
k
k
k
steps to the right and
m
m
m
steps up” or ”
m
m
m
steps to the right and
k
k
k
steps up” (which can be performed in any way). The spider first makes
l
l
l
”movements” and in then, moves to the right or up without any restriction. If
n
=
m
⋅
l
n = m \cdot l
n
=
m
⋅
l
, find all possible ways the spider can approach the point
C
C
C
, where
n
,
m
,
k
,
l
n, m, k, l
n
,
m
,
k
,
l
are positive integers with
k
<
m
k < m
k
<
m
. https://cdn.artofproblemsolving.com/attachments/2/d/4fb71086beb844ca7c492a30c7d333fa08d381.png
C2
1
Hide problems
there exists a rectangle whose vertices are centers of marked squares
Some squares of an
n
×
n
n \times n
n
×
n
chessboard have been marked (
n
∈
N
∗
n \in N^*
n
∈
N
∗
). Prove that if the number of marked squares is at least
n
(
n
+
1
2
)
n\left(\sqrt{n} + \frac12\right)
n
(
n
+
2
1
)
, then there exists a rectangle whose vertices are centers of marked squares.
G5
1
Hide problems
cyclic wanted, 2 circles related
Let
A
B
C
ABC
A
BC
be an acute triangle with
A
B
<
A
C
<
B
C
AB < AC < BC
A
B
<
A
C
<
BC
inscribed in a circle
(
c
)
(c)
(
c
)
and let
E
E
E
be an arbitrary point on its altitude
C
D
CD
C
D
. The circle
(
c
1
)
(c_1)
(
c
1
)
with diameter
E
C
EC
EC
, intersects the circle
(
c
)
(c)
(
c
)
at point
K
K
K
(different than
C
C
C
), the line
A
C
AC
A
C
at point
L
L
L
and the line
B
C
BC
BC
at point
M
M
M
. Finally the line
K
E
KE
K
E
intersects
A
B
AB
A
B
at point
N
N
N
. Prove that the quadrilateral
D
L
M
N
DLMN
D
L
MN
is cyclic.
G4
1
Hide problems
concurrency wanted, 2 circles and tangents related
Let
c
(
O
,
R
)
c(O, R)
c
(
O
,
R
)
be a circle,
A
B
AB
A
B
a diameter and
C
C
C
an arbitrary point on the circle different than
A
A
A
and
B
B
B
such that
∠
A
O
C
>
9
0
o
\angle AOC > 90^o
∠
A
OC
>
9
0
o
. On the radius
O
C
OC
OC
we consider point
K
K
K
and the circle
c
1
(
K
,
K
C
)
c_1(K, KC)
c
1
(
K
,
K
C
)
. The extension of the segment
K
B
KB
K
B
meets the circle
(
c
)
(c)
(
c
)
at point
E
E
E
. From
E
E
E
we consider the tangents
E
S
ES
ES
and
E
T
ET
ET
to the circle
(
c
1
)
(c_1)
(
c
1
)
. Prove that the lines
B
E
,
S
T
BE, ST
BE
,
ST
and
A
C
AC
A
C
are concurrent.
G3
1
Hide problems
parallelogram wanted, intersecting circles and prependiculars given
Two circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
intersect at points
M
,
N
M,N
M
,
N
. A line
ℓ
\ell
ℓ
is tangent to
Γ
1
,
Γ
2
\Gamma_1 ,\Gamma_2
Γ
1
,
Γ
2
at
A
A
A
and
B
B
B
, respectively. The lines passing through
A
A
A
and
B
B
B
and perpendicular to
ℓ
\ell
ℓ
intersects
M
N
MN
MN
at
C
C
C
and
D
D
D
respectively. Prove that
A
B
C
D
ABCD
A
BC
D
is a parallelogram.
G2
1
Hide problems
collinearity wanted, parallelograms given
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral, let
O
O
O
be the intersection point of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
, and let
P
P
P
be the intersection point of sides
A
B
AB
A
B
and
C
D
CD
C
D
. Consider the parallelograms
A
O
D
E
AODE
A
O
D
E
and
B
O
C
F
BOCF
BOCF
. Prove that
E
,
F
E, F
E
,
F
and
P
P
P
are collinear.