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Bundeswettbewerb Mathematik
1981 Bundeswettbewerb Mathematik
1981 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
1
2
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Bundeswettbewerb Mathematik 1981 Problem 1.1
Let
a
a
a
and
n
n
n
be positive integers and
s
=
a
+
a
2
+
⋯
+
a
n
s = a + a^2 + \cdots + a^n
s
=
a
+
a
2
+
⋯
+
a
n
. Prove that the last digit of
s
s
s
is
1
1
1
if and only if the last digits of
a
a
a
and
n
n
n
are both equal to
1
1
1
.
Bundeswettbewerb Mathematik 1981 Problem 2.1
A sequence
a
1
,
a
2
,
a
3
,
…
a_1, a_2, a_3, \ldots
a
1
,
a
2
,
a
3
,
…
is defined as follows:
a
1
a_1
a
1
is a positive integer and
a
n
+
1
=
⌊
3
2
a
n
⌋
+
1
a_{n+1} = \left\lfloor \frac{3}{2} a_n \right\rfloor +1
a
n
+
1
=
⌊
2
3
a
n
⌋
+
1
for all
n
∈
N
n \in \mathbb{N}
n
∈
N
. Can
a
1
a_1
a
1
be chosen in such a way that the first
100000
100000
100000
terms of the sequence are even, but the
100001
100001
100001
-th term is odd?
3
2
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Bundeswettbewerb Mathematik 1981 Problem 1.3
A square of sidelength
2
n
2^n
2
n
is divided into unit squares. One of the unit squares is deleted. Prove that the rest of the square can be tiled with
L
L
L
-trominos.
Mind Blowing Number Theory
Let
n
=
2
k
n = 2^k
n
=
2
k
. Prove that we can select
n
n
n
integers from any
2
n
−
1
2n-1
2
n
−
1
integers such that their sum is divisible by
n
n
n
.
2
2
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Bundeswettbewerb Mathematik 1981 Problem 1.2
Prove that if the sides
a
,
b
,
c
a, b, c
a
,
b
,
c
of a non-equilateral triangle satisfy
a
+
b
=
2
c
a + b = 2c
a
+
b
=
2
c
, then the line passing through the incenter and centroid is parallel to one of the sides of the triangle.
A bijective mapping
A bijective mapping from a plane to itself maps every circle to a circle. Prove that it maps every line to a line.
4
1
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2^p + 3^p=a^n Italian TST problem
Prove that for any prime number
p
p
p
the equation
2
p
+
3
p
=
a
n
2^p+3^p=a^n
2
p
+
3
p
=
a
n
has no solution
(
a
,
n
)
(a,n)
(
a
,
n
)
in integers greater than
1
1
1
.