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National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1988 Bundeswettbewerb Mathematik
1988 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
1
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a_{n+1}= |a_n - b_n|, b_{n+1}= |b_n - c_n|, c_{n+1}= |c_n - d_n|, d_{n+1}=...
Starting with four given integers
a
1
,
b
1
,
c
1
,
d
1
a_1, b_1, c_1, d_1
a
1
,
b
1
,
c
1
,
d
1
is defined recursively for all positive integers
n
n
n
:
a
n
+
1
:
=
∣
a
n
−
b
n
∣
,
b
n
+
1
:
=
∣
b
n
−
c
n
∣
,
c
n
+
1
:
=
∣
c
n
−
d
n
∣
,
d
n
+
1
:
=
∣
d
n
−
a
n
∣
.
a_{n+1} := |a_n - b_n|, b_{n+1} := |b_n - c_n|, c_{n+1} := |c_n - d_n|, d_{n+1} := |d_n - a_n|.
a
n
+
1
:=
∣
a
n
−
b
n
∣
,
b
n
+
1
:=
∣
b
n
−
c
n
∣
,
c
n
+
1
:=
∣
c
n
−
d
n
∣
,
d
n
+
1
:=
∣
d
n
−
a
n
∣.
Prove that there is a natural number
k
k
k
such that all terms
a
k
,
b
k
,
c
k
,
d
k
a_k, b_k, c_k, d_k
a
k
,
b
k
,
c
k
,
d
k
take the value zero.
1
2
Hide problems
n^3 game pieces on n^4 fields of a square
A square is divided into
n
4
n^4
n
4
fields like a chessboard.
n
3
n^3
n
3
game pieces are placed on these squares placed, on each at most one. There are the same number of stones in each row. Besides, the whole arrangement symmetrical to one of the diagonals of the square; this diagonal is called
d
d
d
. Prove that: a) If
n
n
n
is odd, then there is at least one stone on
d
d
d
. b) If
n
n
n
is even, then there is an arrangement of the type described, in which there is no stone on
d
d
d
.
x-y,2x + 2y + 1,3x + 3y + 1 are pefrect squares if 2x^2 + x = 3y^2 + y
For the natural numbers
x
x
x
and
y
y
y
,
2
x
2
+
x
=
3
y
2
+
y
2x^2 + x = 3y^2 + y
2
x
2
+
x
=
3
y
2
+
y
. Prove that then
x
−
y
x-y
x
−
y
,
2
x
+
2
y
+
1
2x + 2y + 1
2
x
+
2
y
+
1
and
3
x
+
3
y
+
1
3x + 3y + 1
3
x
+
3
y
+
1
are perfect squares.
3
2
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octagon [with equal angles and rational sides]
Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.
orthic triangles of same perimeter criterion on acute triangles
Prove that all acute-angled triangles with the equal altitudes
h
c
h_c
h
c
and the equal angles
γ
\gamma
γ
have orthic triangles with same perimeters.
2
2
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heights and inner radius of a triangle (BWM1988_1_2)
Let
h
a
h_a
h
a
,
h
b
h_b
h
b
and
h
c
h_c
h
c
be the heights and
r
r
r
the inradius of a triangle. Prove that the triangle is equilateral if and only if
h
a
+
h
b
+
h
c
=
9
r
h_a + h_b + h_c = 9r
h
a
+
h
b
+
h
c
=
9
r
.
3k points divide a circle into k arcs of lengths 1, 2 and 3
A circle is somehow divided by
3
k
3k
3
k
points into
k
k
k
arcs of lengths
1
,
2
1, 2
1
,
2
and
3
3
3
each. Prove that two of these points are always diametrically opposite.