MathDB
Problems
Contests
International Contests
Balkan MO
2016 Balkan MO
2016 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
Hide problems
Coloring the Squares of an Infinite Grid
The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of
1201
1201
1201
colours so that no rectangle with perimeter
100
100
100
contains two squares of the same colour. Show that no rectangle of size
1
×
1201
1\times1201
1
×
1201
or
1201
×
1
1201\times1
1201
×
1
contains two squares of the same colour.Note: Any rectangle is assumed here to have sides contained in the lines of the grid.(Bulgaria - Nikolay Beluhov)
3
1
Hide problems
Primes Dividing Polynomial
Find all monic polynomials
f
f
f
with integer coefficients satisfying the following condition: there exists a positive integer
N
N
N
such that
p
p
p
divides
2
(
f
(
p
)
!
)
+
1
2(f(p)!)+1
2
(
f
(
p
)!)
+
1
for every prime
p
>
N
p>N
p
>
N
for which
f
(
p
)
f(p)
f
(
p
)
is a positive integer.Note: A monic polynomial has a leading coefficient equal to 1.(Greece - Panagiotis Lolas and Silouanos Brazitikos)
2
1
Hide problems
Cyclic Quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with
A
B
<
C
D
AB<CD
A
B
<
C
D
. The diagonals intersect at the point
F
F
F
and lines
A
D
AD
A
D
and
B
C
BC
BC
intersect at the point
E
E
E
. Let
K
K
K
and
L
L
L
be the orthogonal projections of
F
F
F
onto lines
A
D
AD
A
D
and
B
C
BC
BC
respectively, and let
M
M
M
,
S
S
S
and
T
T
T
be the midpoints of
E
F
EF
EF
,
C
F
CF
CF
and
D
F
DF
D
F
respectively. Prove that the second intersection point of the circumcircles of triangles
M
K
T
MKT
M
K
T
and
M
L
S
MLS
M
L
S
lies on the segment
C
D
CD
C
D
.(Greece - Silouanos Brazitikos)
1
1
Hide problems
Injective Function
Find all injective functions
f
:
R
→
R
f: \mathbb R \rightarrow \mathbb R
f
:
R
→
R
such that for every real number
x
x
x
and every positive integer
n
n
n
,
∣
∑
i
=
1
n
i
(
f
(
x
+
i
+
1
)
−
f
(
f
(
x
+
i
)
)
)
∣
<
2016
\left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016
i
=
1
∑
n
i
(
f
(
x
+
i
+
1
)
−
f
(
f
(
x
+
i
))
)
<
2016
(Macedonia)