MathDB
Problems
Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1995 Bundeswettbewerb Mathematik
1995 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
1
2
Hide problems
game with 2 persons and 2 heaps of p and q stones
A game is played with two heaps of
p
p
p
and
q
q
q
stones. Two players alternate playing, with
A
A
A
starting. A player in turn takes away one heap and divides the other heap into two smaller ones. A player who cannot perform a legal move loses the game. For which values of
p
p
p
and
q
q
q
can
A
A
A
force a victory?
moving stones in lattice points, (a,b) ->(2a,b) or (a,2b), (a-b,b) or (a,b-a)
Starting at
(
1
,
1
)
(1,1)
(
1
,
1
)
, a stone is moved in the coordinate plane according to the following rules: (i) From any point
(
a
,
b
)
(a,b)
(
a
,
b
)
, the stone can move to
(
2
a
,
b
)
(2a,b)
(
2
a
,
b
)
or
(
a
,
2
b
)
(a,2b)
(
a
,
2
b
)
. (ii) From any point
(
a
,
b
)
(a,b)
(
a
,
b
)
, the stone can move to
(
a
−
b
,
b
)
(a-b,b)
(
a
−
b
,
b
)
if
a
>
b
a > b
a
>
b
, or to
(
a
,
b
−
a
)
(a,b-a)
(
a
,
b
−
a
)
if
a
<
b
a < b
a
<
b
. For which positive integers
x
,
y
x,y
x
,
y
can the stone be moved to
(
x
,
y
)
(x,y)
(
x
,
y
)
?
2
2
Hide problems
locus of third vertices of equilateral triangles
A line
g
g
g
and a point
A
A
A
outside
g
g
g
are given in a plane. A point
P
P
P
moves along
g
g
g
. Find the locus of the third vertices of equilateral triangles whose two vertices are
A
A
A
and
P
P
P
.
union of finitely many disjoint subintervals of [0,1], total length <= 1/2
Let
S
S
S
be a union of finitely many disjoint subintervals of
[
0
,
1
]
[0,1]
[
0
,
1
]
such that no two points in
S
S
S
have distance
1
/
10
1/10
1/10
. Show that the total length of the intervals comprising
S
S
S
is at most
1
/
2
1/2
1/2
.
3
2
Hide problems
a+b = n , x/a + y/b = 1
A natural number
n
n
n
is called breakable if there exist positive integers
a
,
b
,
x
,
y
a,b,x,y
a
,
b
,
x
,
y
such that
a
+
b
=
n
a+b = n
a
+
b
=
n
and
x
a
+
y
b
=
1
\frac{x}{a}+\frac{y}{b}= 1
a
x
+
b
y
=
1
. Find all breakable numbers.
each diagonal of convex pentagon is parallel to one side of the pentagon
Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.
4
2
Hide problems
at most 400 unit discs are inside a square of side 100
A number of unit discs are given inside a square of side
100
100
100
such that (i) no two of the discs have a common interior point, and (ii) every segment of length
10
10
10
, lying entirely within the square, meets at least one disc. Prove that there are at least
400
400
400
discs in the square.
multiple of k <= k^4 whose decimal expension has at most 4 distinct digits
Prove that every integer
k
>
1
k > 1
k
>
1
has a multiple less than
k
4
k^4
k
4
whose decimal expension has at most four distinct digits.