MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
1962 German National Olympiad
1962 German National Olympiad
Part of
German National Olympiad
Subcontests
(5)
3
1
Hide problems
rectangular sheets
With a roller shear, rectangular sheets of
1420
1420
1420
mm wide should be made, namely with a width of
500
500
500
mm and a total length of
1000
1000
1000
m as well as a width of
300
300
300
mm and a total length of
1800
1800
1800
m can be cut. So far it has been based on the attached drawing cut, in which the gray area represents the waste, which is quite large. A socialist brigade proposes cutting in such a way that waste is significantly reduced becomes. a) What percentage is the waste if cutting continues as before? b) How does the brigade have to cut so that the waste is as small as possible and what is the total length of the starting sheets is required in this case? c) What percentage is the waste now? https://cdn.artofproblemsolving.com/attachments/f/8/c6c88b79abb5d34674bf54524ae1731985c3f7.png
1
1
Hide problems
y = a - b 10^{-kx}
In 27,000 fertilization trials with phosphorus fertilizers, the following average average crop yields for potatoes were found:
F
e
r
t
i
l
i
z
e
r
a
p
p
l
i
c
a
t
i
o
n
b
a
s
e
d
o
n
P
2
O
5
(
d
t
/
h
a
)
′
c
r
o
p
y
i
e
l
d
(
d
t
/
h
a
)
Fertilizer \,\, application \,\,based \,\,on \,\,P2O5 (dt/ha) \ '\ crop \,\, yield \,\, (dt/ha)
F
er
t
i
l
i
zer
a
ppl
i
c
a
t
i
o
n
ba
se
d
o
n
P
2
O
5
(
d
t
/
ha
)
′
cro
p
y
i
e
l
d
(
d
t
/
ha
)
0.0
237
0.0 \ \ 237
0.0
237
0.3
251
0.3 \ \ 251
0.3
251
0.9
269
0.9 \ \ 269
0.9
269
The relationship between the fertilizer application
x
x
x
(in dt/ha) and the crop yield
y
y
y
(in dt/ha), can be approximated by the following relation:
y
=
a
−
b
⋅
1
0
−
k
x
y = a - b \cdot 10^{-kx}
y
=
a
−
b
⋅
1
0
−
k
x
where
a
,
b
a, b
a
,
b
and
k
k
k
are constants. a) Calculate these constants using the values given above! b) Calculate the crop yield for a fertilizer application of
0.6
0.6
0.6
dt/ha and
1.2
1.2
1.2
dt/ha! c) Set the percentage deviation of the calculated values from those determined in the experiment values
261
261
261
dt/ha or
275
275
275
dt/ha.
2
1
Hide problems
one of u(1 -v), v(1 -w), w(1 - u) is <=1/4 if 0<u,w,v<1
Let
u
,
v
u, v
u
,
v
and
w
w
w
be any positive numbers smaller than
1
1
1
. Prove that among the numbers
u
(
1
−
v
)
u(1 -v)
u
(
1
−
v
)
,
v
(
1
−
w
)
v(1 -w)
v
(
1
−
w
)
,
w
(
1
−
u
)
w(1 - u)
w
(
1
−
u
)
there is always at least one value not greater than
1
4
\frac14
4
1
.
5
1
Hide problems
locus of midpoints in 3D
Given a plane
P
P
P
and two fixed points
A
A
A
and
B
B
B
that do not lie in this plane. Denote two points
A
′
A'
A
′
and
B
′
B'
B
′
on plane
P
P
P
and
M
,
N
M ,N
M
,
N
the midpoints of the segments
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
. a) Determine the locus of the midpoint of the segment MN if the points are
A
′
A'
A
′
and
B
′
B'
B
′
move arbitrarily in plane
P
P
P
. b) A circle
O
O
O
is considered in the plane
P
P
P
. Determine the locus
L
L
L
of the midpoint of the segment
M
N
MN
MN
if the points
A
′
A'
A
′
and
B
′
B'
B
′
lie on the circle
O
O
O
or inside it . c)
A
′
A'
A
′
is assumed to be fixed on the circle
O
O
O
or inside it and
B
′
B'
B
′
is assumed to be movable inside it , except for
O
O
O
. Determine the locus of the point
B
′
B'
B
′
such the above certain locus
L
L
L
remains the same . Note: For b) and c) the following cases should be considered: 1.
A
′
A'
A
′
and
B
′
B'
B
′
are different, 2.
A
′
A'
A
′
and
B
′
B'
B
′
coincide.
4
1
Hide problems
max / min distance >= \sqrt2 in convex quad
A convex flat quadrilateral is given. Prove that for the ratio
q
q
q
of the largest to the smallest of all distances, for any two vertices:
q
≥
2
q \ge \sqrt2
q
≥
2
.[hide=original wording]Gegeben sei ein konvexes ebenes Viereck. Es ist zu beweisen, dass fur den Quotienten q aus dem großten und dem kleinsten aller Abstande zweier beliebiger Eckpunkte voneinander stets gilt: q >= \sqrt2.