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Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2019 Bundeswettbewerb Mathematik
2019 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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Zero digits in the expansion of Sqrt(2)
In the decimal expansion of
2
=
1.4142
…
\sqrt{2}=1.4142\dots
2
=
1.4142
…
, Isabelle finds a sequence of
k
k
k
successive zeroes where
k
k
k
is a positive integer.Show that the first zero of this sequence can occur no earlier than at the
k
k
k
-th position after the decimal point.
between 10k and 10k + 100 there are not more than 23 primes
Prove that for no integer
k
≥
2
k \ge 2
k
≥
2
, between
10
k
10k
10
k
and
10
k
+
100
10k + 100
10
k
+
100
there are more than
23
23
23
prime numbers.
3
2
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Geometry with circle, square and parallel lines
Let
A
B
C
D
ABCD
A
BC
D
be a square. Choose points
E
E
E
on
B
C
BC
BC
and
F
F
F
on
C
D
CD
C
D
so that
∠
E
A
F
=
4
5
∘
\angle EAF=45^\circ
∠
E
A
F
=
4
5
∘
and so that neither
E
E
E
nor
F
F
F
is a vertex of the square. The lines
A
E
AE
A
E
and
A
F
AF
A
F
intersect the circumcircle of the square in the points
G
G
G
and
H
H
H
distinct from
A
A
A
, respectively. Show that the lines
E
F
EF
EF
and
G
H
GH
G
H
are parallel.
<MTW=<TLM wanted, incircle, median, angle bisector, altitude related
Let
A
B
C
ABC
A
BC
be atriangle with
A
C
‾
>
B
C
‾
\overline{AC}> \overline{BC}
A
C
>
BC
and incircle
k
k
k
. Let
M
,
W
,
L
M,W,L
M
,
W
,
L
be the intersections of the median, angle bisector and altitude from point
C
C
C
respectively. The tangent to
k
k
k
passing through
M
M
M
, that is different from
A
B
AB
A
B
, touch
k
k
k
in
T
T
T
. Prove that the angles
∠
M
T
W
\angle MTW
∠
MT
W
and
∠
T
L
M
\angle TLM
∠
T
L
M
are equal.
2
2
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Six digit number divisible by 271
The lettes
A
,
C
,
F
,
H
,
L
A,C,F,H,L
A
,
C
,
F
,
H
,
L
and
S
S
S
represent six not necessarily distinct decimal digits so that
S
≠
0
S \ne 0
S
=
0
and
F
≠
0
F \ne 0
F
=
0
. We form the two six-digit numbers
S
C
H
L
A
F
SCHLAF
SC
H
L
A
F
and
F
L
A
C
H
S
FLACHS
F
L
A
C
H
S
.Show that the difference of these two numbers is divisible by
271
271
271
if and only if
C
=
L
C=L
C
=
L
and
H
=
A
H=A
H
=
A
.Remark: The words "Schlaf" and "Flachs" are German for "sleep" and "flax".
min of ab/c+bc/a+ca/b when a,b,c>0 with a^2 + b^2 + c^2 = 1
Determine the smallest possible value of the sum
S
(
a
,
b
,
c
)
=
a
b
c
+
b
c
a
+
c
a
b
S (a, b, c) = \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}
S
(
a
,
b
,
c
)
=
c
ab
+
a
b
c
+
b
c
a
where
a
,
b
,
c
a, b, c
a
,
b
,
c
are three positive real numbers with
a
2
+
b
2
+
c
2
=
1
a^2 + b^2 + c^2 = 1
a
2
+
b
2
+
c
2
=
1
1
2
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Chessboard covering with dominoes. Is there a square?
An
8
×
8
8 \times 8
8
×
8
chessboard is covered completely and without overlaps by
32
32
32
dominoes of size
1
×
2
1 \times 2
1
×
2
. Show that there are two dominoes forming a
2
×
2
2 \times 2
2
×
2
square.
120 pirates split 119 gold pieces, no more than 14 each
120
120
120
pirates distribute
119
119
119
gold pieces among themselves. Then the captain checks if any pirate has
15
15
15
or more gold pieces. If he finds the first one, he must give all his gold pieces to other pirates, whereby he may not give more than one gold piece to anyone. This control is repeated as long as there is any pirate with
15
15
15
or more gold pieces. Does this process end after a lot of checks?