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1969 German National Olympiad
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Part of
1969 German National Olympiad
Problems
(1)
periodic decimal fractions
Source: 1968-69 Germany R4 12.1
10/13/2024
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.
number theory
algebra