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National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2004 Bundeswettbewerb Mathematik
2004 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
1
2
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Modulo rulez
At the beginning of a game, I write the numbers
1
1
1
,
2
2
2
, ...,
2004
2004
2004
onto a desk. A move consists of - selecting some numbers standing on the desk; - calculating the rest of the sum of these numbers under division by
11
11
11
and writing this rest onto the desk; - deleting the selected numbers. In such a game, after a number of moves, only two numbers remained on the desk. One of them was
1000
1000
1000
. What was the other one?
Easy number theory from the bwm
Let
k
k
k
be a positive integer. A natural number
m
m
m
is called
k
k
k
-typical if each divisor of
m
m
m
leaves the remainder
1
1
1
when being divided by
k
k
k
. Prove: a) If the number of all divisors of a positive integer
n
n
n
(including the divisors
1
1
1
and
n
n
n
) is
k
k
k
-typical, then
n
n
n
is the
k
k
k
-th power of an integer. b) If
k
>
2
k > 2
k
>
2
, then the converse of the assertion a) is not true.
2
2
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A^2 + b^2 + c^2 is a perfect square
Consider a triangle whose sidelengths
a
a
a
,
b
b
b
,
c
c
c
are integers, and which has the property that one of its altitudes equals the sum of the two others. Then, prove that
a
2
+
b
2
+
c
2
a^2+b^2+c^2
a
2
+
b
2
+
c
2
is a perfect square.
Beautiful combinatorial geometry
Let
k
k
k
be a positive integer. In a circle with radius
1
1
1
, finitely many chords are drawn. You know that every diameter of the circle intersects at most
k
k
k
of these chords. Prove that the sum of the lengths of all these chords is less than
k
⋅
π
k \cdot \pi
k
⋅
π
.
3
2
Hide problems
Two congruent regular hexagons
Prove that two congruent regular hexagons can be cut up into (altogether)
6
6
6
parts such that these
6
6
6
parts can be composed to form an equilateral triangle (without gaps or overlaps).
Triangle geometry, bisecting chords
Given two circles
k
1
k_1
k
1
and
k
2
k_2
k
2
which intersect at two different points
A
A
A
and
B
B
B
. The tangent to the circle
k
2
k_2
k
2
at the point
A
A
A
meets the circle
k
1
k_1
k
1
again at the point
C
1
C_1
C
1
. The tangent to the circle
k
1
k_1
k
1
at the point
A
A
A
meets the circle
k
2
k_2
k
2
again at the point
C
2
C_2
C
2
. Finally, let the line
C
1
C
2
C_1C_2
C
1
C
2
meet the circle
k
1
k_1
k
1
in a point
D
D
D
different from
C
1
C_1
C
1
and
B
B
B
. Prove that the line
B
D
BD
B
D
bisects the chord
A
C
2
AC_2
A
C
2
.
4
2
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a cube is decomposed in a finite number of parallelepipeds
A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.
Sqrt(x^2 + y^3) and sqrt(x^3 + y^2) are rational
Prove that there exist infinitely many pairs
(
x
;
y
)
\left(x;\;y\right)
(
x
;
y
)
of different positive rational numbers, such that the numbers
x
2
+
y
3
\sqrt{x^2+y^3}
x
2
+
y
3
and
x
3
+
y
2
\sqrt{x^3+y^2}
x
3
+
y
2
are both rational.