MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1991 Federal Competition For Advanced Students
1991 Federal Competition For Advanced Students
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
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circle
Let
A
B
AB
A
B
be a chord of a circle
k
k
k
of radius
r
r
r
, with AB\equal{}c.
(
a
)
(a)
(
a
)
Construct the triangle
A
B
C
ABC
A
BC
with
C
C
C
on
k
k
k
in which a median from
A
A
A
or
B
B
B
is of a given length
d
.
d.
d
.
(
b
)
(b)
(
b
)
For which
c
c
c
and
d
d
d
is this triangle unique?
3
1
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find the number of squares
Find the number of squares in the sequence given by a_0\equal{}91 and a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n for
n
≥
0.
n \ge 0.
n
≥
0.
2
1
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solve the equation
Solve in real numbers the equation: \frac{1}{x}\plus{}\frac{1}{x\plus{}2}\minus{}\frac{1}{x\plus{}4}\minus{}\frac{1}{x\plus{}6}\minus{}\frac{1}{x\plus{}8}\minus{}\frac{1}{x\plus{}10}\plus{}\frac{1}{x\plus{}12}\plus{}\frac{1}{x\plus{}14}\equal{}0.
1
1
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old exercise
Suppose that
a
,
b
,
a,b,
a
,
b
,
and \sqrt[3]{a}\plus{}\sqrt[3]{b} are rational numbers. Prove that
a
3
\sqrt[3]{a}
3
a
and
b
3
\sqrt[3]{b}
3
b
are also rational.