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Bundeswettbewerb Mathematik
1975 Bundeswettbewerb Mathematik
1975 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
1
2
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sequence of lattice points with non-negative integer coordinates
In a planar coordinate system, the points have non-negative integer coordinates numbered according to the figure. E.g. the point
(
3
,
1
)
(3,1)
(
3
,
1
)
has the number
12
12
12
. https://cdn.artofproblemsolving.com/attachments/a/a/28725d75f281ac4618129067037d751c8d8f83.png What is the number of the point
(
x
,
y
)
(x,y)
(
x
,
y
)
?
Bundeswettbewerb Mathematik 1975 Problem 2.1
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be distinct positive real numbers. Prove that if one of the numbers
c
,
d
c, d
c
,
d
lies between
a
a
a
and
b
b
b
, or one of
a
,
b
a, b
a
,
b
lies between
c
c
c
and
d
d
d
, then
(
a
+
b
)
(
c
+
d
)
>
a
b
+
c
d
\sqrt{(a+b)(c+d)} >\sqrt{ab} +\sqrt{cd}
(
a
+
b
)
(
c
+
d
)
>
ab
+
c
d
and that otherwise, one can choose
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
so that this inequality is false.
4
2
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Bundeswettbewerb Mathematik 1975 Problem 1.4
In the country of Sikinia there are finitely many cities. From each city, exactly three roads go out and each road goes to another Sikinian city. A tourist starts a trip from city
A
A
A
and drives according to the following rule: he turns left at the first city, then right at the next city, and so on, alternately. Show that he will eventually return to
A
.
A.
A
.
2 brothers inherited n gold pieces of the total weight 2n
Two brothers inherited
n
n
n
gold pieces of the total weight
2
n
2n
2
n
. The weights of the pieces are integers, and the heaviest piece is not heavier than all the other pieces together. Show that if
n
n
n
is even, the brother can divide the inheritance into two parts of equal weight.
3
2
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Bundeswettbewerb Mathematik 1975 Problem 1.3
Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.
a_n/g_n is a positive integer, then x_1 = x_2 = ··· = x_n
For
n
n
n
positive integers
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x2,...,x_n
x
1
,
x
2
,
...
,
x
n
,
a
n
a_n
a
n
is their arithmetic and
g
n
g_n
g
n
the geometric mean. Consider the statement
S
n
S_n
S
n
: If
a
n
/
g
n
a_n/g_n
a
n
/
g
n
is a positive integer, then
x
1
=
x
2
=
⋅
⋅
⋅
=
x
n
x_1 = x_2 = ··· = x_n
x
1
=
x
2
=
⋅⋅⋅
=
x
n
. Prove
S
2
S_2
S
2
and disprove
S
n
S_n
S
n
for all even
n
>
2
n > 2
n
>
2
.
2
2
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in each polyhedron there exist two faces with the same number of edges
Prove that in each polyhedron there exist two faces with the same number of edges.
no prime in sequence 10001, 100010001, 1000100010001, ...
Prove that no term of the sequence
10001
10001
10001
,
100010001
100010001
100010001
,
1000100010001
1000100010001
1000100010001
,
.
.
.
...
...
is prime.