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Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2014 Bundeswettbewerb Mathematik
2014 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
2
2
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Vertices of a prism are labeled with numbers 1 to 100
The
100
100
100
vertices of a prism, whose base is a
50
50
50
-gon, are labeled with numbers
1
,
2
,
3
,
…
,
100
1, 2, 3, \ldots, 100
1
,
2
,
3
,
…
,
100
in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than
48
48
48
. Note: In all the triangles the three vertices do not lie on a straight line.
Find all sequences that satisfy the following equation
For all positive integers
m
m
m
and
k
k
k
with
m
≥
k
m\ge k
m
≥
k
, define
a
m
,
k
=
(
m
k
−
1
)
−
3
m
−
k
a_{m,k}=\binom{m}{k-1}-3^{m-k}
a
m
,
k
=
(
k
−
1
m
)
−
3
m
−
k
. Determine all sequences of real numbers
{
x
1
,
x
2
,
x
3
,
…
}
\{x_1, x_2, x_3, \ldots\}
{
x
1
,
x
2
,
x
3
,
…
}
, such that each positive integer
n
n
n
satisfies the equation
a
n
,
1
x
1
+
a
n
,
2
x
2
+
⋯
+
a
n
,
n
x
n
=
0
a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0
a
n
,
1
x
1
+
a
n
,
2
x
2
+
⋯
+
a
n
,
n
x
n
=
0
1
2
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BWM 2014 P1: Mean of any 3 numbers is in the board
Anja has to write
2014
2014
2014
integers on the board such that arithmetic mean of any of the three numbers is among those
2014
2014
2014
numbers. Show that this is possible only when she writes nothing but
2014
2014
2014
equal integers.
3^(n+1) | 2^(3^n)+1: BWM 2014: P5
Show that for all positive integers
n
n
n
, the number
2
3
n
+
1
2^{3^n}+1
2
3
n
+
1
is divisible by
3
n
+
1
3^{n+1}
3
n
+
1
.
3
2
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BWM 2014 P3: Construct with a ruler, given a regular hexagon
A regular hexagon with side length
1
1
1
is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is
7
\sqrt7
7
.Notes: ''Using a ruler construct points
…
\ldots
…
'' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.
Find the max number of points of intersection of semicircles
A line
g
g
g
is given in a plane.
n
n
n
distinct points are chosen arbitrarily from
g
g
g
and are named as
A
1
,
A
2
,
…
,
A
n
A_1, A_2, \ldots, A_n
A
1
,
A
2
,
…
,
A
n
. For each pair of points
A
i
,
A
j
A_i,A_j
A
i
,
A
j
, a semicircle is drawn with
A
i
A_i
A
i
and
A
j
A_j
A
j
as its endpoints. All semicircles lie on the same side of
g
g
g
. Determine the maximum number of points (which are not lying in
g
g
g
) of intersection of semicircles as a function of
n
n
n
.
4
2
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Terminating decimal expansion
Find all postive integers
n
n
n
for which the number
4
n
+
1
n
(
2
n
−
1
)
\frac{4n+1}{n(2n-1)}
n
(
2
n
−
1
)
4
n
+
1
has a terminating decimal expansion.
A_n is the centroid of A_{n-1}A_{n-2}A_{n-3}
Three non-collinear points
A
1
,
A
2
,
A
3
A_1, A_2, A_3
A
1
,
A
2
,
A
3
are given in a plane. For
n
=
4
,
5
,
6
,
…
n = 4, 5, 6, \ldots
n
=
4
,
5
,
6
,
…
,
A
n
A_n
A
n
be the centroid of the triangle
A
n
−
3
A
n
−
2
A
n
−
1
A_{n-3}A_{n-2}A_{n-1}
A
n
−
3
A
n
−
2
A
n
−
1
. a) Show that there is exactly one point
S
S
S
, which lies in the interior of the triangle
A
n
−
3
A
n
−
2
A
n
−
1
A_{n-3}A_{n-2}A_{n-1}
A
n
−
3
A
n
−
2
A
n
−
1
for all
n
≥
4
n\ge 4
n
≥
4
. b) Let
T
T
T
be the intersection of the line
A
1
A
2
A_1A_2
A
1
A
2
with
S
A
3
SA_3
S
A
3
. Determine the two ratios,
A
1
T
:
T
A
2
A_1T : TA_2
A
1
T
:
T
A
2
and
T
S
:
S
A
3
TS : SA_3
TS
:
S
A
3
.