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National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2000 Bundeswettbewerb Mathematik
2000 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(6)
1b
1
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2 numbes with same 2000 digits and have both sum 999 ... 999
Two natural numbers have the same decimal digits in different order and have the sum
999
⋯
999
999\cdots 999
999
⋯
999
. Is this possible when each of the numbers consists of
2000
2000
2000
digits?
1a
1
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Two natural numbers have the same decimal digits in diffe...
Two natural numbers have the same decimal digits in different order and have the sum
999
⋯
999
999\cdots 999
999
⋯
999
. Is this possible when each of the numbers consists of
1999
1999
1999
digits?
4
2
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Can the sum be equal to zero? [Germany BW 2000]
Consider the sums of the form
∑
k
=
1
n
ϵ
k
k
3
,
\sum_{k=1}^{n} \epsilon_k k^3,
∑
k
=
1
n
ϵ
k
k
3
,
where
ϵ
k
∈
{
−
1
,
1
}
.
\epsilon_k \in \{-1, 1\}.
ϵ
k
∈
{
−
1
,
1
}
.
Is any of these sums equal to
0
0
0
if(a)
n
=
2000
;
n=2000;
n
=
2000
;
(b)
n
=
2001
?
n=2001 \ ?
n
=
2001
?
a circular game board is divided into n sectors, empty or occupied by marker
A circular game board is divided into
n
≥
3
n \ge 3
n
≥
3
sectors. Each sector is either empty or occupied by a marker. In each step one chooses an occupied sector, removes its marker and then switches each of the two adjacent sectors from occupied to empty or vice-versa. Starting with a single occupied sector, for which
n
n
n
is it possible to end up with all empty sectors after finitely many steps?
2
2
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a+b is divisible by a-b [Germany BW 2000]
Prove that for every integer
n
≥
2
n \geq 2
n
≥
2
there exist
n
n
n
different positive integers such that for any two of these integers
a
a
a
and
b
b
b
their sum
a
+
b
a+b
a
+
b
is divisible by their difference
a
−
b
.
a - b.
a
−
b
.
5-tuple with property sum of any 3 is divisible by the sum of the other 2
A
5
5
5
-tuple
(
1
,
1
,
1
,
1
,
2
)
(1,1,1,1,2)
(
1
,
1
,
1
,
1
,
2
)
has the property that the sum of any three of them is divisible by the sum of the remaining two. Is there a
5
5
5
-tuple with this property whose all terms are distinct?
1
1
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We can divide weights to 3 groups [Germany BW 2000]
We are given
n
≥
3
n \geq 3
n
≥
3
weights of masses
1
,
2
,
3
,
…
,
n
1, 2, 3, \ldots , n
1
,
2
,
3
,
…
,
n
grams. Find all
n
n
n
for which it is possible to divide these weights into three groups with the same mass.
3
2
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Four spheres pass through a single point [Germany BW 2000]
For each vertex of a given tetrahedron, a sphere passing through that vertex and the midpoints of the edges outgoing from this vertex is constructed. Prove that these four spheres pass through a single point.
bisection in quad
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a semicircle with diameter
A
B
AB
A
B
. The diagonals
A
C
,
B
D
AC,BD
A
C
,
B
D
intersect at
S
S
S
, and
T
T
T
is the projection of
S
S
S
on
A
B
AB
A
B
. Show that
S
T
ST
ST
bisects angle
C
T
D
CTD
CT
D
.