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Contests
International Contests
Danube Competition in Mathematics
2018 Danube Mathematical Competition
2018 Danube Mathematical Competition
Part of
Danube Competition in Mathematics
Subcontests
(4)
1
2
Hide problems
d_i are all the proper divisors of n, d_i+1 are all the proper divisors of m
Find all the pairs
(
n
,
m
)
(n, m)
(
n
,
m
)
of positive integers which fulfil simultaneously the conditions: i) the number
n
n
n
is composite; ii) if the numbers
d
1
,
d
2
,
.
.
.
,
d
k
,
k
∈
N
∗
d_1, d_2, ..., d_k, k \in N^*
d
1
,
d
2
,
...
,
d
k
,
k
∈
N
∗
are all the proper divisors of
n
n
n
, then the numbers
d
1
+
1
,
d
2
+
1
,
.
.
.
,
d
k
+
1
d_1 + 1, d_2 + 1, . . . , d_k + 1
d
1
+
1
,
d
2
+
1
,
...
,
d
k
+
1
are all the proper divisors of
m
m
m
.
\sum_{i=1}^{k} x_i \le k - 1, inequality with labeling beads from necklace
Suppose we have a necklace of
n
n
n
beads. Each bead is labeled with an integer and the sum of all these labels is
n
−
1
n - 1
n
−
1
. Prove that we can cut the necklace to form a string, whose consecutive labels
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
satisfy
∑
i
=
1
k
x
i
≤
k
−
1
\sum_{i=1}^{k} x_i \le k - 1
∑
i
=
1
k
x
i
≤
k
−
1
for any
k
=
1
,
.
.
.
,
n
k = 1,...,n
k
=
1
,
...
,
n
3
2
Hide problems
a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n , a_i \in Q
Find all the positive integers
n
n
n
with the property: there exists an integer
k
>
2
k > 2
k
>
2
and the positive rational numbers
a
1
,
a
2
,
.
.
.
,
a
k
a_1, a_2, ..., a_k
a
1
,
a
2
,
...
,
a
k
such that
a
1
+
a
2
+
.
.
+
a
k
=
a
1
a
2
.
.
.
a
k
=
n
a_1 + a_2 + .. + a_k = a_1a_2 . . . a_k = n
a
1
+
a
2
+
..
+
a
k
=
a
1
a
2
...
a
k
=
n
.
danube senior parallel wanted 2018 P3
Let
A
B
C
ABC
A
BC
be an acute non isosceles triangle. The angle bisector of angle
A
A
A
meets again the circumcircle of the triangle
A
B
C
ABC
A
BC
in
D
D
D
. Let
O
O
O
be the circumcenter of the triangle
A
B
C
ABC
A
BC
. The angle bisectors of
∠
A
O
B
\angle AOB
∠
A
OB
, and
∠
A
O
C
\angle AOC
∠
A
OC
meet the circle
γ
\gamma
γ
of diameter
A
D
AD
A
D
in
P
P
P
and
Q
Q
Q
respectively. The line
P
Q
PQ
PQ
meets the perpendicular bisector of
A
D
AD
A
D
in
R
R
R
. Prove that
A
R
/
/
B
C
AR // BC
A
R
//
BC
.
2
2
Hide problems
danube junior angle chasing 2018 P2
Let
A
B
C
ABC
A
BC
be a triangle such that in its interior there exists a point
D
D
D
with
∠
D
A
C
=
∠
D
C
A
=
3
0
o
\angle DAC = \angle DCA = 30^o
∠
D
A
C
=
∠
D
C
A
=
3
0
o
and
∠
D
B
A
=
6
0
o
\angle DBA = 60^o
∠
D
B
A
=
6
0
o
. Denote
E
E
E
the midpoint of the segment
B
C
BC
BC
, and take
F
F
F
on the segment
A
C
AC
A
C
so that
A
F
=
2
F
C
AF = 2FC
A
F
=
2
FC
. Prove that
D
E
⊥
E
F
DE \perp EF
D
E
⊥
EF
.
infinite pairs of (m, n) such m \ n^2+1 and n \m^2+1 ,
Prove that there are in finitely many pairs of positive integers
(
m
,
n
)
(m, n)
(
m
,
n
)
such that simultaneously
m
m
m
divides
n
2
+
1
n^2 + 1
n
2
+
1
and
n
n
n
divides
m
2
+
1
m^2 + 1
m
2
+
1
.
4
2
Hide problems
number of the subsets of M whose sum of elements equals n
Let
M
M
M
be the set of positive odd integers. For every positive integer
n
n
n
, denote
A
(
n
)
A(n)
A
(
n
)
the number of the subsets of
M
M
M
whose sum of elements equals
n
n
n
. For instance,
A
(
9
)
=
2
A(9) = 2
A
(
9
)
=
2
, because there are exactly two subsets of
M
M
M
with the sum of their elements equal to
9
9
9
:
{
9
}
\{9\}
{
9
}
and
{
1
,
3
,
5
}
\{1, 3, 5\}
{
1
,
3
,
5
}
. a) Prove that
A
(
n
)
≤
A
(
n
+
1
)
A(n) \le A(n + 1)
A
(
n
)
≤
A
(
n
+
1
)
for every integer
n
≥
2
n \ge 2
n
≥
2
. b) Find all the integers
n
≥
2
n \ge 2
n
≥
2
such that
A
(
n
)
=
A
(
n
+
1
)
A(n) = A(n + 1)
A
(
n
)
=
A
(
n
+
1
)
Disconnected Chessboard Coloring
Let
n
≥
3
n \geq 3
n
≥
3
be an odd number and suppose that each square in a
n
×
n
n \times n
n
×
n
chessboard is colored either black or white. Two squares are considered adjacent if they are of the same color and share a common vertex and two squares
a
,
b
a,b
a
,
b
are considered connected if there exists a sequence of squares
c
1
,
…
,
c
k
c_1,\ldots,c_k
c
1
,
…
,
c
k
with
c
1
=
a
,
c
k
=
b
c_1 = a, c_k = b
c
1
=
a
,
c
k
=
b
such that
c
i
,
c
i
+
1
c_i, c_{i+1}
c
i
,
c
i
+
1
are adjacent for
i
=
1
,
2
,
…
,
k
−
1
i=1,2,\ldots,k-1
i
=
1
,
2
,
…
,
k
−
1
. \\ \\ Find the maximal number
M
M
M
such that there exists a coloring admitting
M
M
M
pairwise disconnected squares.