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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1984 Czech And Slovak Olympiad IIIA
1984 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
3
1
Hide problems
a_{n + 2} =4a_{n + 1}-3a_n, b_n= [ a_{n+1} / a_{n-1} ]
Let the sequence
{
a
n
}
\{a_n\}
{
a
n
}
,
n
≥
0
n \ge 0
n
≥
0
satisfy the recurrence relation
a
n
+
2
=
4
a
n
+
1
−
3
a
n
,
(
1
)
a_{n + 2} =4a_{n + 1}-3a_n, \ \ (1)
a
n
+
2
=
4
a
n
+
1
−
3
a
n
,
(
1
)
Let us define the sequence
{
b
n
}
\{b_n\}
{
b
n
}
,
n
≥
1
n \ge 1
n
≥
1
by the relation
b
n
=
[
a
n
+
1
a
n
−
1
]
b_n= \left[ \frac{a_{n+1}}{a_{n-1}} \right]
b
n
=
[
a
n
−
1
a
n
+
1
]
where we put
b
n
=
1
b_n =1
b
n
=
1
for
a
n
−
1
=
0
a_{n-1}=0
a
n
−
1
=
0
. Prove that, starting from a certain term, the sequence also satisfies the recurrence relation (1). Note:
[
x
]
[x]
[
x
]
indicates the whole part of the number
x
x
x
.
6
1
Hide problems
f(f(m)) =-m
Let f be a function from the set Z of all integers into itself, that satisfies the condition for all
m
∈
Z
m \in Z
m
∈
Z
,
f
(
f
(
m
)
)
=
−
m
.
(
1
)
f(f(m)) =-m. \ \ (1)
f
(
f
(
m
))
=
−
m
.
(
1
)
Then: (a)
f
f
f
is a mutually unique mapping, i.e. a simple mapping of the set
Z
Z
Z
onto the set
Z
Z
Z
, (b) for all
m
∈
Z
m \in Z
m
∈
Z
holds that
f
(
−
m
)
=
−
f
(
m
)
f(-m) = -f(m)
f
(
−
m
)
=
−
f
(
m
)
, (c)
f
(
m
)
=
0
f(m) = 0
f
(
m
)
=
0
if and only if
m
=
0
m = 0
m
=
0
. Prove these statements and construct an example of a mapping f that satisfies condition (1).
5
1
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convex polyhedron, n edges, one vertex with four edfes and each other with 3
Find all natural numbers
n
n
n
for which there exists a convex polyhedron with
n
n
n
edges, with exactly one vertex having four edges and all other vertices having
3
3
3
edges.
4
1
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x^y = r where x,y irrationals and r natural >1
Let
r
r
r
be a natural number greater than
1
1
1
. Then there exist positive irrational numbers
x
,
y
x, y
x
,
y
such that
x
y
=
r
x^y = r
x
y
=
r
. Prove it.
2
1
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cosA+cosB+cosC+cosD =0 => ABCD is cyclic or trapezoid
Let
α
,
β
,
γ
,
δ
\alpha, \beta, \gamma, \delta
α
,
β
,
γ
,
δ
be the interior angles of a convex quadrilateral, If
cos
α
+
cos
β
+
cos
γ
,
+
cos
δ
=
0
,
\cos\alpha + \cos\beta + \cos\gamma, + \cos\delta = 0 ,
cos
α
+
cos
β
+
cos
γ
,
+
cos
δ
=
0
,
then this quadrilateral is cyclic or a trapezium. Prove it.
1
1
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exista a plane intersecting n rays, cube related
A cube
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A_1A_2A_3A_4A_5A_6A_7A_8
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
is given in space. We will mark its center with the letter
S
S
S
(intersection of solid diagonals). Find all natural numbers
k
k
k
for which there exists a plane not containing the point
S
S
S
and intersecting just
k
k
k
of the rays
S
A
1
,
S
A
2
,
.
.
S
A
8
SA_1, SA_2, .. SA_8
S
A
1
,
S
A
2
,
..
S
A
8