MathDB
Problems
Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2003 Bundeswettbewerb Mathematik
2003 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
3
1
Hide problems
Nice geometry from MMO
Consider a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
, and let
S
S
S
be the intersection of
A
C
AC
A
C
and
B
D
BD
B
D
. Let
E
E
E
and
F
F
F
the orthogonal projections of
S
S
S
on
A
B
AB
A
B
and
C
D
CD
C
D
respectively. Prove that the perpendicular bisector of segment
E
F
EF
EF
meets the segments
A
D
AD
A
D
and
B
C
BC
BC
at their midpoints.
2
2
Hide problems
X^3 - 4x^2 - 16x + 60 = y, ...
Find all triples
(
x
,
y
,
z
)
\left(x,\ y,\ z\right)
(
x
,
y
,
z
)
of integers satisfying the following system of equations:
x
3
−
4
x
2
−
16
x
+
60
=
y
x^3-4x^2-16x+60=y
x
3
−
4
x
2
−
16
x
+
60
=
y
;
y
3
−
4
y
2
−
16
y
+
60
=
z
y^3-4y^2-16y+60=z
y
3
−
4
y
2
−
16
y
+
60
=
z
;
z
3
−
4
z
2
−
16
z
+
60
=
x
z^3-4z^2-16z+60=x
z
3
−
4
z
2
−
16
z
+
60
=
x
.
all elements of the sequence are integers
The sequence
{
a
1
,
a
2
,
…
}
\{a_1,a_2,\ldots\}
{
a
1
,
a
2
,
…
}
is recursively defined by
a
1
=
1
a_1 = 1
a
1
=
1
,
a
2
=
1
a_2 = 1
a
2
=
1
,
a
3
=
2
a_3 = 2
a
3
=
2
, and
a
n
+
3
=
1
a
n
⋅
(
a
n
+
1
a
n
+
2
+
7
)
,
∀
n
>
0.
a_{n+3} = \frac 1{a_n}\cdot (a_{n+1}a_{n+2}+7), \ \forall \ n > 0.
a
n
+
3
=
a
n
1
⋅
(
a
n
+
1
a
n
+
2
+
7
)
,
∀
n
>
0.
Prove that all elements of the sequence are integers.
1
2
Hide problems
Six consecutive positive integers
Given six consecutive positive integers, prove that there exists a prime such that one and only one of these six integers is divisible by this prime.
sum of a linear and periodic funtion
The graph of a function
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
has two has at least two centres of symmetry. Prove that
f
f
f
can be represented as sum of a linear and periodic funtion.
4
2
Hide problems
BWM2003
Determine all positive integers which cannot be represented as
a
b
+
a
+
1
b
+
1
\frac{a}{b}+\frac{a+1}{b+1}
b
a
+
b
+
1
a
+
1
with
a
,
b
a,b
a
,
b
being positive integers.
numbers z, z+p and z+q
Let
p
p
p
and
q
q
q
be two positive integers that have no common divisor. The set of integers shall be partioned into three subsets
A
A
A
,
B
B
B
,
C
C
C
such that for each integer
z
z
z
in each of the sets
A
A
A
,
B
B
B
,
C
C
C
there is exactly one of the numbers
z
z
z
,
z
+
p
z+p
z
+
p
and
z
+
q
z+q
z
+
q
. a) Prove that such a decomposition is possible if and only if
p
+
q
p+q
p
+
q
is divisible by
3
3
3
. b) In the case we omit the restriction that
p
p
p
,
q
q
q
may not have a common divisor, prove that for
p
≠
q
p \neq q
p
=
q
the number
p
+
q
gcd
(
p
,
q
)
\frac{p+q}{\gcd(p,q)}
g
c
d
(
p
,
q
)
p
+
q
is divisible by 3.