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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
1969 German National Olympiad
1969 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
3
1
Hide problems
u * v = \dfrac{u + v}{1 + \dfrac{uv}{c^2}} semigroup
A set
M
M
M
of elements
u
,
v
,
w
u, v, w
u
,
v
,
w
is called a semigroup if an operation is defined in it is which uniquely assigns an element
w
w
w
from
M
M
M
to every ordered pair
(
u
,
v
)
(u, v)
(
u
,
v
)
of elements from
M
M
M
(you write
u
⊗
v
=
w
u \otimes v = w
u
⊗
v
=
w
) and if this algebraic operation is associative, i.e. if for all elements
u
,
v
,
w
u, v,w
u
,
v
,
w
from
M
M
M
:
(
u
⊗
v
)
⊗
w
=
u
⊗
(
v
⊗
w
)
.
(u \otimes v) \otimes w = u \otimes (v \otimes w).
(
u
⊗
v
)
⊗
w
=
u
⊗
(
v
⊗
w
)
.
Now let
c
c
c
be a positive real number and let
M
M
M
be the set of all non-negative real numbers that are smaller than
c
c
c
. For each two numbers
u
,
v
u, v
u
,
v
from
M
M
M
we define:
u
⊗
v
=
u
+
v
1
+
u
v
c
2
u \otimes v = \dfrac{u + v}{1 + \dfrac{uv}{c^2}}
u
⊗
v
=
1
+
c
2
uv
u
+
v
Investigate a) whether
M
M
M
is a semigroup; b) whether this semigroup is regular, i.e. whether from
u
⊗
v
1
=
u
⊗
v
2
u \otimes v_1 = u\otimes v_2
u
⊗
v
1
=
u
⊗
v
2
always
v
1
=
v
2
v_1 = v_2
v
1
=
v
2
and from
v
1
⊗
u
=
v
2
⊗
u
v_1 \otimes u = v_2 \otimes u
v
1
⊗
u
=
v
2
⊗
u
also
v
1
=
v
2
v_1 = v_2
v
1
=
v
2
follows.
1
1
Hide problems
periodic decimal fractions
Every nonnegative periodic decimal fraction represents a rational number, also in the form
p
q
\frac{p}{q}
q
p
can be represented (
p
p
p
and
q
q
q
are natural numbers and coprime,
p
≥
0
p\ge 0
p
≥
0
,
q
>
0
)
q > 0)
q
>
0
)
. Now let
a
1
a_1
a
1
,
a
2
a_2
a
2
,
a
3
a_3
a
3
and
a
4
a_4
a
4
be digits to represent numbers in the decadic system. Let
a
1
≠
a
3
a_1 \ne a_3
a
1
=
a
3
or
a
2
≠
a
4
a_2 \ne a_4
a
2
=
a
4
.Prove that it for the numbers:
z
1
=
0
,
a
1
a
2
a
3
a
4
‾
=
0
,
a
1
a
2
a
3
a
4
a
1
a
2
a
3
a
4
.
.
.
z_1 = 0, \overline{a_1a_2a_3a_4} = 0,a_1a_2a_3a_4a_1a_2a_3a_4...
z
1
=
0
,
a
1
a
2
a
3
a
4
=
0
,
a
1
a
2
a
3
a
4
a
1
a
2
a
3
a
4
...
z
2
=
0
,
a
4
a
1
a
2
a
3
‾
z_2 = 0, \overline{a_4a_1a_2a_3}
z
2
=
0
,
a
4
a
1
a
2
a
3
z
3
=
0
,
a
3
a
4
a
1
a
2
‾
z_3 = 0, \overline{a_3a_4a_1a_2}
z
3
=
0
,
a
3
a
4
a
1
a
2
z
4
=
0
,
a
2
a
3
a
4
a
1
‾
z_4 = 0, \overline{a_2a_3a_4a_1}
z
4
=
0
,
a
2
a
3
a
4
a
1
In the above representation
p
/
q
p/q
p
/
q
always have the same denominator.[hide=original wording]Jeder nichtnegative periodische Dezimalbruch repr¨asentiert eine rationale Zahl, die auch in der Form p/q dargestellt werden kann (p und q nat¨urliche Zahlen und teilerfremd, p >= 0, q > 0). Nun seien a1, a2, a3 und a4 Ziffern zur Darstellung von Zahlen im dekadischen System. Dabei sei a1
≠
\ne
=
a3 oder a2
≠
\ne
=
a4. Beweisen Sie! Die Zahlen z1 = 0, a1a2a3a4 = 0,a1a2a3a4a1a2a3a4... z2 = 0, a4a1a2a3 z3 = 0, a3a4a1a2 z4 = 0, a2a3a4a1 haben in der obigen Darstellung p/q stets gleiche Nenner.
6
1
Hide problems
F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x) has no roots
Let
n
n
n
be a positive integer,
h
h
h
a real number and
f
(
x
)
f(x)
f
(
x
)
a polynomial (whole rational function) with real coefficients of degree n, which has no real zeros. Prove that then also the polynomial
F
(
x
)
=
f
(
x
)
+
h
f
′
(
x
)
+
h
2
f
′
′
(
x
)
+
.
.
.
+
h
n
f
(
n
)
(
x
)
F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x)
F
(
x
)
=
f
(
x
)
+
h
f
′
(
x
)
+
h
2
f
′′
(
x
)
+
...
+
h
n
f
(
n
)
(
x
)
has no real zeros.
5
1
Hide problems
sin 5x = 16 sin x sin (x -\pi/5) sin (x +\pi/5) sin (x -2\pi/5) sin (x +2\pi/5)
Prove that for all real numbers
x
x
x
holds:
sin
5
x
=
16
sin
x
⋅
sin
(
x
−
π
5
)
⋅
sin
(
x
−
2
π
5
)
sin
(
x
+
2
π
5
)
\sin 5x = 16 \sin x \cdot \sin \left(x -\frac{\pi}{5} \right) \cdot \sin\left(x -\frac{2\pi}{5} \right) \sin \left(x +\frac{2\pi}{5} \right)
sin
5
x
=
16
sin
x
⋅
sin
(
x
−
5
π
)
⋅
sin
(
x
−
5
2
π
)
sin
(
x
+
5
2
π
)
4
1
Hide problems
|log_2(x + y)| + | log_2(x - y)| = 3, xy = 3
Solve the system of equations:
∣
log
2
(
x
+
y
)
∣
+
∣
log
2
(
x
−
y
)
∣
=
3
|\log_2(x + y)| + | \log_2(x - y)| = 3
∣
lo
g
2
(
x
+
y
)
∣
+
∣
lo
g
2
(
x
−
y
)
∣
=
3
x
y
=
3
xy = 3
x
y
=
3
2
1
Hide problems
inversion constructions
There is a circle
k
k
k
in a plane with center
M
M
M
and radius
r
r
r
. The following illustration, through which every point
P
≠
M
P \ne M
P
=
M
., is called a “reflection on the circle
k
k
k
” from
ε
\varepsilon
ε
a point
P
′
P'
P
′
from
ε
\varepsilon
ε
is assigned:(1)
P
′
P'
P
′
lies on the ray emanating from
M
M
M
and passing through
P
P
P
. (2) It is
M
P
⋅
M
P
′
=
r
2
MP \cdot MP' = r^2
MP
⋅
M
P
′
=
r
2
.a) Construct the mirror point
P
′
P'
P
′
for any given point
P
≠
M
P \ne M
P
=
M
inside
k
k
k
.b) Let another circle
k
1
k_1
k
1
be given arbitrarily, but such that
M
M
M
lies outside
k
1
k_1
k
1
.Construct
k
1
′
k'_1
k
1
′
, i.e. the set of all mirror points
P
′
P'
P
′
of the points
P
P
P
of
k
1
k_1
k
1
.