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Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2020 Bundeswettbewerb Mathematik
2020 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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Find the prime factorization of recursive sequence involving maximum
Define a sequence
(
a
n
)
(a_n)
(
a
n
)
recursively by
a
1
=
0
,
a
2
=
2
,
a
3
=
3
a_1=0, a_2=2, a_3=3
a
1
=
0
,
a
2
=
2
,
a
3
=
3
and
a
n
=
max
0
<
d
<
n
a
d
⋅
a
n
−
d
a_n=\max_{0<d<n} a_d \cdot a_{n-d}
a
n
=
max
0
<
d
<
n
a
d
⋅
a
n
−
d
for
n
≥
4
n \ge 4
n
≥
4
. Determine the prime factorization of
a
19702020
a_{19702020}
a
19702020
.
Non-negative numbers in a table, row sum larger than column sum
In each cell of a table with
m
m
m
rows and
n
n
n
columns, where
m
<
n
m<n
m
<
n
, we put a non-negative real number such that each column contains at least one positive number.Show that there is a cell with a positive number such that the sum of the numbers in its row is larger than the sum of the numbers in its column.
3
2
Hide problems
Sum of product of distances is constant
Let
A
B
AB
A
B
be the diameter of a circle
k
k
k
and let
E
E
E
be a point in the interior of
k
k
k
. The line
A
E
AE
A
E
intersects
k
k
k
a second time in
C
≠
A
C \ne A
C
=
A
and the line
B
E
BE
BE
intersects
k
k
k
a second time in
D
≠
B
D \ne B
D
=
B
.Show that the value of
A
C
⋅
A
E
+
B
D
⋅
B
E
AC \cdot AE+BD\cdot BE
A
C
⋅
A
E
+
B
D
⋅
BE
is independent of the choice of
E
E
E
.
Points moving on two lines with constant speed, common point of circles
Two lines
m
m
m
and
n
n
n
intersect in a unique point
P
P
P
. A point
M
M
M
moves along
m
m
m
with constant speed, while another point
N
N
N
moves along
n
n
n
with the same speed. They both pass through the point
P
P
P
, but not at the same time.Show that there is a fixed point
Q
≠
P
Q \ne P
Q
=
P
such that the points
P
,
Q
,
M
P,Q,M
P
,
Q
,
M
and
N
N
N
lie on a common circle all the time.
2
2
Hide problems
Moving a knight in the minimal number of moves
Konstantin moves a knight on a
n
×
n
n \times n
n
×
n
- chess board from the lower left corner to the lower right corner with the minimal number of moves.Then Isabelle takes the knight and moves it from the lower left corner to the upper right corner with the minimal number of moves.For which values of
n
n
n
do they need the same number of moves?
No rational solutions to a system of equations
Prove that there are no rational numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
with
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
and
x
2
+
y
2
+
z
2
=
100
x^2+y^2+z^2=100
x
2
+
y
2
+
z
2
=
100
.
1
2
Hide problems
Putting quite a bit of gold in two treasure chests
Leo and Smilla find
2020
2020
2020
gold nuggets with masses
1
,
2
,
…
,
2020
1,2,\dots,2020
1
,
2
,
…
,
2020
gram, which they distribute to a red and a blue treasure chest according to the following rule:First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest.This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest.How many gram of gold can Smilla make sure to win?
Perfect squares as sums or differences of powers
Show that there are infinitely many perfect squares of the form
5
0
m
−
5
0
n
50^m-50^n
5
0
m
−
5
0
n
, but no perfect square of the form
202
0
m
+
202
0
n
2020^m+2020^n
202
0
m
+
202
0
n
, where
m
m
m
and
n
n
n
are positive integers.