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Bundeswettbewerb Mathematik
1977 Bundeswettbewerb Mathematik
1977 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
3
2
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Bundeswettbewerb Mathematik 1977 Problem 1.3
The number
50
50
50
is written as a sum of several positive integers (not necessarily distinct) whose product is divisible by
100.
100.
100.
What is the largest possible value of this product?
Bundeswettbewerb Mathematik 1977 Problem 2.3
Show that there are infinitely many positive integers
a
a
a
that cannot be written as
a
=
a
1
6
+
a
2
6
+
…
+
a
7
6
,
a = a_{1}^{6}+ a_{2}^{6} + \ldots + a_{7}^{6},
a
=
a
1
6
+
a
2
6
+
…
+
a
7
6
,
where the
a
i
a_i
a
i
are positive integers. State and prove a generalization.
2
2
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Bundeswettbewerb Mathematik 1977 Problem 2.2
On a plane are given three non-collinear points
A
,
B
,
C
A, B, C
A
,
B
,
C
. We are given a disk of diameter different from that of the circle passing through
A
,
B
,
C
A, B, C
A
,
B
,
C
large enough to cover all three points. Construct the fourth vertex of the parallelogram
A
B
C
D
ABCD
A
BC
D
using only this disk (The disk is to be used as a circular ruler, for constructing a circle passing through two given points).
Bundeswettbewerb Mathematik 1977 Problem 1.2
A beetle crawls along the edges of an
n
n
n
-lateral pyramid, starting and ending at the midpoint
A
A
A
of a base edge and passing through each point at most once. How many ways are there for the beetle to do this (two ways are said to be equal if they go through the same vertices)? Show that the sum of the numbers of passed vertices (over all these ways) equals
1
2
+
2
2
+
…
+
n
2
.
1^2 +2^2 +\ldots +n^2.
1
2
+
2
2
+
…
+
n
2
.
4
1
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All functions f with f(x)+f(1- 1/x)=x
Find all functions
f
:
R
→
R
f : \mathbb R \to \mathbb R
f
:
R
→
R
such that
f
(
x
)
+
f
(
1
−
1
x
)
=
x
,
f(x)+f\left(1-\frac{1}{x}\right)=x,
f
(
x
)
+
f
(
1
−
x
1
)
=
x
,
holds for all real
x
x
x
.
1
2
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Bundeswettbewerb Mathematik 1977 Problem 1.1
Among
2000
2000
2000
distinct positive integers, there are equally many even and odd ones. The sum of the numbers is less than
3000000.
3000000.
3000000.
Show that at least one of the numbers is divisible by
3.
3.
3.
every natural number sum of a + b
Does there exist two infinite sets
A
,
B
A,B
A
,
B
such that every number can be written uniquely as a sum of an element of
A
A
A
and an element of
B
B
B
?