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Problems
Contests
International Contests
Danube Competition in Mathematics
2014 Danube Mathematical Competition
2014 Danube Mathematical Competition
Part of
Danube Competition in Mathematics
Subcontests
(4)
2
2
Hide problems
at least 806 palindrome words to stick for a palindrome of 2014 letters
We call word a sequence of letters
l
1
l
2
.
.
.
l
n
‾
,
n
≥
1
\overline {l_1l_2...l_n}, n\ge 1
l
1
l
2
...
l
n
,
n
≥
1
. A word
l
1
l
2
.
.
.
l
n
‾
,
n
≥
1
\overline {l_1l_2...l_n}, n\ge 1
l
1
l
2
...
l
n
,
n
≥
1
is called palindrome if
l
k
=
l
n
−
k
+
1
l_k=l_{n-k+1}
l
k
=
l
n
−
k
+
1
, for any
k
,
1
≤
k
≤
n
k, 1 \le k \le n
k
,
1
≤
k
≤
n
. Consider a word
X
=
l
1
l
2
.
.
.
l
2014
‾
X=\overline {l_1l_2...l_{2014}}
X
=
l
1
l
2
...
l
2014
in which
l
k
∈
{
A
,
B
}
l_k\in\{A,B\}
l
k
∈
{
A
,
B
}
, for any
k
,
1
≤
k
≤
2014
k, 1\le k \le 2014
k
,
1
≤
k
≤
2014
. Prove that there are at least
806
806
806
palindrome words to ''stick" together to get word
X
X
X
.
set with \lfloor x\rfloor =\lfloor y\rfloor , for any x,y
Let
S
S
S
be a set of positive integers such that
⌊
x
⌋
=
⌊
y
⌋
\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor
⌊
x
⌋
=
⌊
y
⌋
for all
x
,
y
∈
S
x, y \in S
x
,
y
∈
S
. Show that the products
x
y
xy
x
y
, where
x
,
y
∈
S
x, y \in S
x
,
y
∈
S
, are pairwise distinct.
4
2
Hide problems
\Sum a_1=0 from i=1,...,2n, among all pairs 2n-1 have sum >=0
Consider the real numbers
a
1
,
a
2
,
.
.
.
,
a
2
n
a_1,a_2,...,a_{2n}
a
1
,
a
2
,
...
,
a
2
n
whose sum is equal to
0
0
0
. Prove that among pairs
(
a
i
,
a
j
)
,
i
<
j
(a_i,a_j) , i<j
(
a
i
,
a
j
)
,
i
<
j
where
i
,
j
∈
{
1
,
2
,
.
.
.
,
2
n
}
i,j \in \{1,2,...,2n\}
i
,
j
∈
{
1
,
2
,
...
,
2
n
}
.there are at least
2
n
−
1
2n-1
2
n
−
1
pairs with the property that
a
i
+
a
j
≥
0
a_i+a_j\ge 0
a
i
+
a
j
≥
0
.
max cardinality of set of lattice point with lines no parallel to given triangle
Let
n
n
n
be a positive integer and let
△
\triangle
△
be the closed triangular domain with vertices at the lattice points
(
0
,
0
)
,
(
n
,
0
)
(0, 0), (n, 0)
(
0
,
0
)
,
(
n
,
0
)
and
(
0
,
n
)
(0, n)
(
0
,
n
)
. Determine the maximal cardinality a set
S
S
S
of lattice points in
△
\triangle
△
may have, if the line through every pair of distinct points in
S
S
S
is parallel to no side of
△
\triangle
△
.
1
2
Hide problems
natural a =(p+q)/r+(q+r)/p+(r+p)/q where p, q,r primes
Determine the natural number
a
=
p
+
q
r
+
q
+
r
p
+
r
+
p
q
a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}
a
=
r
p
+
q
+
p
q
+
r
+
q
r
+
p
where
p
,
q
p, q
p
,
q
and
r
r
r
are prime positive numbers.
2 intersecting circles, tangents, intersections, equal segments, tangent wanted
Two circles
γ
1
\gamma_1
γ
1
and
γ
2
\gamma_2
γ
2
cross one another at two points; let
A
A
A
be one of these points. The tangent to
γ
1
\gamma_1
γ
1
at
A
A
A
meets again
γ
2
\gamma_2
γ
2
at
B
B
B
, the tangent to
γ
2
\gamma_2
γ
2
at
A
A
A
meets again
γ
1
\gamma_1
γ
1
at
C
C
C
, and the line
B
C
BC
BC
meets again
γ
1
\gamma_1
γ
1
and
γ
2
\gamma_2
γ
2
at
D
1
D_1
D
1
and
D
2
D_2
D
2
, respectively. Let
E
1
E_1
E
1
and
E
2
E_2
E
2
be interior points of the segments
A
D
1
AD_1
A
D
1
and
A
D
2
AD_2
A
D
2
, respectively, such that
A
E
1
=
A
E
2
AE_1 = AE_2
A
E
1
=
A
E
2
. The lines
B
E
1
BE_1
B
E
1
and
A
C
AC
A
C
meet at
M
M
M
, the lines
C
E
2
CE_2
C
E
2
and
A
B
AB
A
B
meet at
N
N
N
, and the lines
M
N
MN
MN
and
B
C
BC
BC
meet at
P
P
P
. Show that the line
P
A
PA
P
A
is tangent to the circle
A
B
C
ABC
A
BC
.
3
2
Hide problems
angle chasing , danube junior geometry 2014
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
<
9
0
o
,
A
B
≠
A
C
\angle A<90^o, AB \ne AC
∠
A
<
9
0
o
,
A
B
=
A
C
. Denote
H
H
H
the orthocenter of triangle
A
B
C
ABC
A
BC
,
N
N
N
the midpoint of segment
[
A
H
]
[AH]
[
A
H
]
,
M
M
M
the midpoint of segment
[
B
C
]
[BC]
[
BC
]
and
D
D
D
the intersection point of the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
with the segment
[
M
N
]
[MN]
[
MN
]
. Prove that
<
A
D
H
=
9
0
o
<ADH=90^o
<
A
DH
=
9
0
o
exist n pairwise coprime composite integers form arithmetic progression
Given any integer
n
≥
2
n \ge 2
n
≥
2
, show that there exists a set of
n
n
n
pairwise coprime composite integers in arithmetic progression.