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Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2016 Bundeswettbewerb Mathematik
2016 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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Two children with the same name in one class
There are
33
33
33
children in a given class. Each child writes a number on the blackboard, which indicates how many other children possess the same forename as oneself. Afterwards, each child does the same thing with their surname. After they've finished, each of the numbers
0
,
1
,
2
,
…
,
10
0,1,2,\dots,10
0
,
1
,
2
,
…
,
10
appear at least once on the blackboard. Prove that there are at least two children in this class that have the same forename and surname.
Dodecahedron's Planes cuts Space in Regions
Each side face of a dodecahedron lies in a uniquely determined plane. Those planes cut the space in a finite number of disjoint regions. Find the number of such regions.
3
2
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Perpendicular lines due to circle
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
and
D
D
D
be points on a circle in this order. The chords
A
C
AC
A
C
and
B
D
BD
B
D
intersect in point
P
P
P
. The perpendicular to
A
C
AC
A
C
through C and the perpendicular to
B
D
BD
B
D
through
D
D
D
intersect in point
Q
Q
Q
. Prove that the lines
A
B
AB
A
B
and
P
Q
PQ
PQ
are perpendicular.
Just like an old Putnam FE...
Find all functions
f
f
f
that is defined on all reals but
1
3
\tfrac13
3
1
and
−
1
3
- \tfrac13
−
3
1
and satisfies
f
(
x
+
1
1
−
3
x
)
+
f
(
x
)
=
x
f \left(\frac{x+1}{1-3x} \right) + f(x) = x
f
(
1
−
3
x
x
+
1
)
+
f
(
x
)
=
x
for all
x
∈
R
∖
{
±
1
3
}
x \in \mathbb{R} \setminus \{ \pm \tfrac13 \}
x
∈
R
∖
{
±
3
1
}
.
2
2
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Geometric game: Maximizing area
A triangle
A
B
C
ABC
A
BC
with area
1
1
1
is given. Anja and Bernd are playing the following game: Anja chooses a point
X
X
X
on side
B
C
BC
BC
. Then Bernd chooses a point
Y
Y
Y
on side
C
A
CA
C
A
und at last Anja chooses a point
Z
Z
Z
on side
A
B
AB
A
B
. Also,
X
,
Y
X,Y
X
,
Y
and
Z
Z
Z
cannot be a vertex of triangle
A
B
C
ABC
A
BC
. Anja wants to maximize the area of triangle
X
Y
Z
XYZ
X
Y
Z
and Bernd wants to minimize that area. What is the area of triangle
X
Y
Z
XYZ
X
Y
Z
at the end of the game, if both play optimally?
Sum of Triangular Number and Prime Number
Prove that there are infinitely many positive integers that cannot be expressed as the sum of a triangular number and a prime number.
1
2
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101010...0101 is not prime
A number with
2016
2016
2016
zeros that is written as
101010
…
0101
101010 \dots 0101
101010
…
0101
is given, in which the zeros and ones alternate. Prove that this number is not prime.
Distinct sums and consecutive numbers
There are
n
(
n
+
1
)
2
\tfrac{n(n+1)}{2}
2
n
(
n
+
1
)
distinct sums of two distinct numbers, if there are
n
n
n
numbers. For which
n
(
n
≥
3
)
n \ (n \geq 3)
n
(
n
≥
3
)
do there exist
n
n
n
distinct integers, such that those sums are
n
(
n
−
1
)
2
\tfrac{n(n-1)}{2}
2
n
(
n
−
1
)
consecutive numbers?