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Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2006 Bundeswettbewerb Mathematik
2006 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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Cut a piece of paper
A piece of paper with the shape of a square lies on the desk. It gets dissected step by step into smaller pieces: in every step, one piece is taken from the desk and cut into two pieces by a straight cut; these pieces are put back on the desk then. Find the smallest number of cuts needed to get
100
100
100
20
20
20
-gons.
digit-reduced numbers
A positive integer is called digit-reduced if at most nine different digits occur in its decimal representation (leading
0
0
0
s are omitted.) Let
M
M
M
be a finite set of digit-reduced numbers. Show that the sum of the reciprocals of the elements in
M
M
M
is less than
180
180
180
.
3
2
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A²+b² > 5c²
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the sidelengths of a triangle such that
a
2
+
b
2
>
5
c
2
a^2+b^2 > 5c^2
a
2
+
b
2
>
5
c
2
holds. Prove that
c
c
c
is the shortest side of the triangle.
locus of point with equal angles
A point
P
P
P
is given inside an acute-angled triangle
A
B
C
ABC
A
BC
. Let
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be the orthogonal projections of
P
P
P
on sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
respectively. Determine the locus of points
P
P
P
for which
∠
B
A
C
=
∠
B
′
A
′
C
′
\angle BAC = \angle B'A'C'
∠
B
A
C
=
∠
B
′
A
′
C
′
and
∠
C
B
A
=
∠
C
′
B
′
A
′
\angle CBA = \angle C'B'A'
∠
CB
A
=
∠
C
′
B
′
A
′
2
2
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x³+y³=4·(x²y+xy²+1)
Prove that there are no integers
x
,
y
x,y
x
,
y
for that it is
x
3
+
y
3
=
4
⋅
(
x
2
y
+
x
y
2
+
1
)
x^3+y^3=4\cdot(x^2y+xy^2+1)
x
3
+
y
3
=
4
⋅
(
x
2
y
+
x
y
2
+
1
)
.
functional eq from positive rationals
Find all functions
f
:
Q
+
→
R
f: Q^{+}\rightarrow R
f
:
Q
+
→
R
such that
f
(
x
)
+
f
(
y
)
+
2
x
y
f
(
x
y
)
=
f
(
x
y
)
f
(
x
+
y
)
f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}
f
(
x
)
+
f
(
y
)
+
2
x
y
f
(
x
y
)
=
f
(
x
+
y
)
f
(
x
y
)
for all
x
,
y
∈
Q
+
x,y\in Q^{+}
x
,
y
∈
Q
+
1
1
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Sums of digits divisible by 2006
Find two consecutive integers with the property that the sums of their digits are each divisible by
2006
2006
2006
.