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Germany Contests
Bundeswettbewerb Mathematik
2024 Bundeswettbewerb Mathematik
2024 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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Tying unit cubes to a thread and building a cuboid
For positive integers
p
p
p
,
q
q
q
and
r
r
r
we are given
p
⋅
q
⋅
r
p \cdot q \cdot r
p
⋅
q
⋅
r
unit cubes. We drill a hole along the space diagonal of each of these cubes and then tie them to a very thin thread of length
p
⋅
q
⋅
r
⋅
3
p \cdot q \cdot r \cdot \sqrt{3}
p
⋅
q
⋅
r
⋅
3
like a string of pearls.We now want to construct a cuboid of side lengths
p
p
p
,
q
q
q
and
r
r
r
out of the cubes, without tearing the thread. a) For which numbers
p
p
p
,
q
q
q
and
r
r
r
is this possible? b) For which numbers
p
p
p
,
q
q
q
and
r
r
r
is this possible in a way such that both ends of the thread coincide?
Welcome back to Sikinia! Have a good flight...
In Sikinia, there are
2024
2024
2024
cities. Between some of them there are flight connections, which can be used in either direction. No city has a direct flight to all
2023
2023
2023
other cities. It is known, however, that there is a positive integer
n
n
n
with the following property: For any
n
n
n
cities in Sikinia, there is another city which is directly connected to all these cities. Determine the largest possible value of
n
n
n
.
3
2
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Circumcircles in a parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram whose diagonals intersect in
M
M
M
. Suppose that the circumcircle of
A
B
M
ABM
A
BM
intersects the segment
A
D
AD
A
D
in a point
E
≠
A
E \ne A
E
=
A
and that the circumcircle of
E
M
D
EMD
EM
D
intersects the segment
B
E
BE
BE
in a point
F
≠
E
F \ne E
F
=
E
. Show that
∠
A
C
B
=
∠
D
C
F
\angle ACB=\angle DCF
∠
A
CB
=
∠
D
CF
.
Bound sum of ratios of areas in triangle
Let
A
B
C
ABC
A
BC
be a triangle. For a point
P
P
P
in its interior, we draw the threee lines through
P
P
P
parallel to the sides of the triangle. This partitions
A
B
C
ABC
A
BC
in three triangles and three quadrilaterals.Let
V
A
V_A
V
A
be the area of the quadrilateral which has
A
A
A
as one vertex. Let
D
A
D_A
D
A
be the area of the triangle which has a part of
B
C
BC
BC
as one of its sides. Define
V
B
,
D
B
V_B, D_B
V
B
,
D
B
and
V
C
,
D
C
V_C, D_C
V
C
,
D
C
similarly.Determine all possible values of
D
A
V
A
+
D
B
V
B
+
D
C
V
C
\frac{D_A}{V_A}+\frac{D_B}{V_B}+\frac{D_C}{V_C}
V
A
D
A
+
V
B
D
B
+
V
C
D
C
, as
P
P
P
varies in the interior of the triangle.
2
2
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Can 44...441 be a perfect square?
Can a number of the form
44
…
41
44\dots 41
44
…
41
, with an odd number of decimal digits
4
4
4
followed by a digit
1
1
1
, be a perfect square?
How slowly can a sequence with no power-of-two sums grow?
Determine the set of all real numbers
r
r
r
for which there exists an infinite sequence
a
1
,
a
2
,
…
a_1,a_2,\dots
a
1
,
a
2
,
…
of positive integers satisfying the following three properties: (1) No number occurs more than once in the sequence. (2) The sum of two different elements of the sequence is never a power of two. (3) For all positive integers
n
n
n
, we have
a
n
<
r
⋅
n
a_n<r \cdot n
a
n
<
r
⋅
n
.
1
2
Hide problems
Solving (x+2)^4-x^4=y^3 over the integers
Determine all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of integers satisfying
(
x
+
2
)
4
−
x
4
=
y
3
.
(x+2)^4-x^4=y^3.
(
x
+
2
)
4
−
x
4
=
y
3
.
Two-player game on a 7x7 board
Arthur and Renate play a game on a
7
×
7
7 \times 7
7
×
7
board. Arthur has two red tiles, initially placed on the cells in the bottom left and the upper right corner. Renate has two black tiles, initially placed on the cells in the bottom right and the upper left corner. In a move, a player can choose one of his two tiles and move them to a horizontally or vertically adjacent cell. The players alternate, with Arthur beginning. Arthur wins when both of his tiles are in horizontally or vertically adjacent cells after some number of moves. Can Renate prevent him from winning?