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National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2021 Bundeswettbewerb Mathematik
2021 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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Colouring line segments in a pyramid
Consider a pyramid with a regular
n
n
n
-gon as its base. We colour all the segments connecting two of the vertices of the pyramid except for the sides of the base either red or blue. Show that if
n
=
9
n=9
n
=
9
then for each such colouring there are three vertices of the pyramid connecting by three segments of the same colour, and that this is not necessarily the case if
n
=
8
n=8
n
=
8
.
Taming a wild rectangle
In the Cartesian plane, a line segment is called tame if it lies parallel to one of the coordinate axes and its distance to this axis is an integer. Otherwise it is called wild.Let
m
m
m
and
n
n
n
be odd positive integers. The rectangle with vertices
(
0
,
0
)
,
(
m
,
0
)
,
(
m
,
n
)
(0,0),(m,0),(m,n)
(
0
,
0
)
,
(
m
,
0
)
,
(
m
,
n
)
and
(
0
,
n
)
(0,n)
(
0
,
n
)
is partitioned into finitely many triangles. Let
M
M
M
be the set of these triangles. Assume that(1) Each triangle from
M
M
M
has at least one tame side. (2) For each tame side of a triangle from
M
M
M
, the corresponding altitude has length
1
1
1
. (3) Each wild side of a triangle from
M
M
M
is a common side of exactly two triangles from
M
M
M
.Show that at least two triangles from
M
M
M
have two tame sides each.
2
2
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Writing 3/n as sum of two reciprocals in exactly 2021 ways
The fraction
3
10
\frac{3}{10}
10
3
can be written as a sum of two reciprocals in exactly two ways:
3
10
=
1
5
+
1
10
=
1
4
+
1
20
\frac{3}{10}=\frac{1}{5}+\frac{1}{10}=\frac{1}{4}+\frac{1}{20}
10
3
=
5
1
+
10
1
=
4
1
+
20
1
a) In how many ways can
3
2021
\frac{3}{2021}
2021
3
be written as a sum of two reciprocals? b) Is there a positive integer
n
n
n
not divisible by
3
3
3
with the property that
3
n
\frac{3}{n}
n
3
can be written as a sum of two reciprocals in exactly
2021
2021
2021
ways?
Looking for a 4-cycle in a graph
A school has 2021 students, each of which knows at least 45 of the other students (where "knowing" is mutual).Show that there are four students who can be seated at a round table such that each of them knows both of her neighbours.
3
2
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Angle between points of intersections of angular bisectors with opposite sides
Consider a triangle
A
B
C
ABC
A
BC
with
∠
A
C
B
=
12
0
∘
\angle ACB=120^\circ
∠
A
CB
=
12
0
∘
. Let
A
’
,
B
’
,
C
’
A’, B’, C’
A
’
,
B
’
,
C
’
be the points of intersection of the angular bisector through
A
A
A
,
B
B
B
and
C
C
C
with the opposite side, respectively. Determine
∠
A
’
C
’
B
’
\angle A’C’B’
∠
A
’
C
’
B
’
.
A classical geometry problem
We are given a circle
k
k
k
and a point
A
A
A
outside of
k
k
k
. Next we draw three lines through
A
A
A
: one secant intersecting the circle
k
k
k
at points
B
B
B
and
C
C
C
, and two tangents touching the circle
k
k
k
at points
D
D
D
and
E
E
E
. Let
F
F
F
be the midpoint of
D
E
DE
D
E
.Show that the line
D
E
DE
D
E
bisects the angle
∠
B
F
C
\angle BFC
∠
BFC
.
1
2
Hide problems
Dividing a cube into three cuboids with integral side lengths
A cube with side length
10
10
10
is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?
Each natural number has a multiple with same sum of digits as its square
Let
Q
(
n
)
Q(n)
Q
(
n
)
denote the sum of the digits of
n
n
n
in its decimal representation. Prove that for every positive integer
k
k
k
, there exists a multiple
n
n
n
of
k
k
k
such that
Q
(
n
)
=
Q
(
n
2
)
Q(n)=Q(n^2)
Q
(
n
)
=
Q
(
n
2
)
.