MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1989 Federal Competition For Advanced Students, P2
1989 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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functions
Determine all functions
f
:
N
0
→
N
0
f: \mathbb{N}_0 \rightarrow \mathbb{N}_0
f
:
N
0
→
N
0
such that f(f(n))\plus{}f(n)\equal{}2n\plus{}6 for all
n
∈
N
0
n \in \mathbb{N}_0
n
∈
N
0
.
5
1
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real solutions
Find all real solutions of the system: x^2\plus{}2yz\equal{}x, y^2\plus{}2zx\equal{}y, z^2\plus{}2xy\equal{}z.
4
1
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maximum area
We are given a circle
k
k
k
and nonparallel tangents
t
1
,
t
2
t_1,t_2
t
1
,
t
2
at points
P
1
,
P
2
P_1,P_2
P
1
,
P
2
on
k
k
k
, respectively. Lines
t
1
t_1
t
1
and
t
2
t_2
t
2
meet at
A
0
A_0
A
0
. For a point
A
3
A_3
A
3
on the smaller arc
P
1
P
2
,
P_1 P_2,
P
1
P
2
,
the tangent
t
3
t_3
t
3
to
k
k
k
at
P
3
P_3
P
3
meets
t
1
t_1
t
1
at
A
1
A_1
A
1
and
t
2
t_2
t
2
at
A
2
A_2
A
2
. How must
P
3
P_3
P
3
be chosen so that the triangle
A
0
A
1
A
2
A_0 A_1 A_2
A
0
A
1
A
2
has maximum area?
3
1
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configurations
Show that it is possible to situate eight parallel planes at equal distances such that each plane contains precisely one vertex of a given cube. How many such configurations of planes are there?
2
1
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triples of integers
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of integers with abc\equal{}1989 and a\plus{}b\minus{}c\equal{}89.
1
1
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real members
Consider the set
S
n
S_n
S
n
of all the
2
n
2^n
2
n
numbers of the type
2
±
2
±
2
±
.
.
.
,
2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},
2
±
2
±
2
±
...
,
where number
2
2
2
appears n\plus{}1 times.
(
a
)
(a)
(
a
)
Show that all members of
S
n
S_n
S
n
are real.
(
b
)
(b)
(
b
)
Find the product
P
n
P_n
P
n
of the elements of
S
n
S_n
S
n
.