MathDB
Problems
Contests
International Contests
Danube Competition in Mathematics
2007 Danube Mathematical Competition
2007 Danube Mathematical Competition
Part of
Danube Competition in Mathematics
Subcontests
(4)
4
1
Hide problems
Danube Mathematical Competition 2007 Problem 4
Let
a
,
n
a,n
a
,
n
be positive integers such that a\ge(n\minus{}1)!. Prove that there exist
n
n
n
distinct prime numbers
p
1
,
…
,
p
n
p_1,\ldots,p_n
p
1
,
…
,
p
n
so that p_i|a\plus{}i, for all i\equal{}\overline{1,\ldots,n}.
3
1
Hide problems
Danube Mathematical Competition 2007 Problem 3
For each positive integer
n
n
n
, define
f
(
n
)
f(n)
f
(
n
)
as the exponent of the
2
2
2
in the decomposition in prime factors of the number
n
!
n!
n
!
. Prove that the equation n\minus{}f(n)\equal{}a has infinitely many solutions for any positive integer
a
a
a
.
2
1
Hide problems
Danube Mathematical Competition 2007 Problem 2
Let
A
B
C
D
ABCD
A
BC
D
be an inscribed quadrilateral and let
E
E
E
be the midpoint of the diagonal
B
D
BD
B
D
. Let
Γ
1
,
Γ
2
,
Γ
3
,
Γ
4
\Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4
Γ
1
,
Γ
2
,
Γ
3
,
Γ
4
be the circumcircles of triangles
A
E
B
AEB
A
EB
,
B
E
C
BEC
BEC
,
C
E
D
CED
CE
D
and
D
E
A
DEA
D
E
A
respectively. Prove that if
Γ
4
\Gamma_4
Γ
4
is tangent to the line
C
D
CD
C
D
, then
Γ
1
,
Γ
2
,
Γ
3
\Gamma_1,\Gamma_2,\Gamma_3
Γ
1
,
Γ
2
,
Γ
3
are tangent to the lines
B
C
,
A
B
,
A
D
BC,AB,AD
BC
,
A
B
,
A
D
respectively.
1
1
Hide problems
Danube Mathematical Competition 2007 Problem 1
Let
n
≥
2
n\ge2
n
≥
2
be a positive integer and denote by
S
n
S_n
S
n
the set of all permutations of the set
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
. For
σ
∈
S
n
\sigma\in S_n
σ
∈
S
n
define
l
(
σ
)
l(\sigma)
l
(
σ
)
to be \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|. Determine
max
σ
∈
S
n
l
(
σ
)
\displaystyle\max_{\sigma\in S_n}l(\sigma)
σ
∈
S
n
max
l
(
σ
)
.