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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1972 Czech and Slovak Olympiad III A
1972 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Construction of points
Two different points
A
,
S
A,S
A
,
S
are given in the plane. Furthermore, positive numbers
d
,
ω
d,\omega
d
,
ω
are given,
ω
<
18
0
∘
.
\omega<180^\circ.
ω
<
18
0
∘
.
Let
X
X
X
be a point and
X
′
X'
X
′
its image under the rotation by the angle
ω
\omega
ω
(in counter-clockwise direction) with respect to the origin
S
.
S.
S
.
Construct all points
X
X
X
such that
X
X
′
=
d
XX'=d
X
X
′
=
d
and
A
A
A
is a point of the segment
X
X
′
.
XX'.
X
X
′
.
Discuss conditions of solvability (in terms of
d
,
ω
,
S
A
d,\omega,SA
d
,
ω
,
S
A
).
5
1
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Disjoint subsets of an n-element set
Determine how many unordered pairs
{
A
,
B
}
\{A,B\}
{
A
,
B
}
is there such that
A
,
B
⊆
{
1
,
…
,
n
}
A,B\subseteq\{1,\ldots,n\}
A
,
B
⊆
{
1
,
…
,
n
}
and
A
∩
B
=
∅
.
A\cap B=\emptyset.
A
∩
B
=
∅.
4
1
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Factorization of $n^4+a$
Show that there are infinitely many positive integers
a
a
a
such that the number
n
4
+
a
n^4+a
n
4
+
a
is composite for every positive integer
n
.
n.
n
.
Give 5 (different) numbers
a
a
a
with the mentioned property.
3
1
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Sequence of polynomials with recursion
Consider a sequence of polynomials such that
P
0
(
x
)
=
2
,
P
1
(
x
)
=
x
P_0(x)=2,P_1(x)=x
P
0
(
x
)
=
2
,
P
1
(
x
)
=
x
and for all
n
≥
1
n\ge1
n
≥
1
P
n
+
1
(
x
)
+
P
n
−
1
(
x
)
=
x
P
n
(
x
)
.
P_{n+1}(x)+P_{n-1}(x)=xP_n(x).
P
n
+
1
(
x
)
+
P
n
−
1
(
x
)
=
x
P
n
(
x
)
.
a) Determine the polynomial
Q
n
(
x
)
=
P
n
2
(
x
)
−
x
P
n
(
x
)
P
n
−
1
(
x
)
+
P
n
−
1
2
(
x
)
Q_n(x)=P^2_n(x)-xP_n(x)P_{n-1}(x)+P^2_{n-1}(x)
Q
n
(
x
)
=
P
n
2
(
x
)
−
x
P
n
(
x
)
P
n
−
1
(
x
)
+
P
n
−
1
2
(
x
)
for
n
=
1972.
n=1972.
n
=
1972.
b) Express the polynomial
(
P
n
+
1
(
x
)
−
P
n
−
1
(
x
)
)
2
\bigl(P_{n+1}(x)-P_{n-1}(x)\bigr)^2
(
P
n
+
1
(
x
)
−
P
n
−
1
(
x
)
)
2
in terms of
P
n
(
x
)
,
Q
n
(
x
)
.
P_n(x),Q_n(x).
P
n
(
x
)
,
Q
n
(
x
)
.
2
1
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Locus on the surface of a cube
Let
A
B
C
D
A
′
B
′
C
′
D
′
ABCDA'B'C'D'
A
BC
D
A
′
B
′
C
′
D
′
be a cube (where
A
B
C
D
ABCD
A
BC
D
is a square and
A
A
′
∥
B
B
′
∥
C
C
′
∥
D
D
′
AA'\parallel BB'\parallel CC'\parallel DD'
A
A
′
∥
B
B
′
∥
C
C
′
∥
D
D
′
). Furthermore, let
R
\mathcal R
R
be a rotation (with respect some line) that maps vertex
A
A
A
to
B
.
B.
B
.
Find the set of all images
X
=
R
(
C
)
X=\mathcal R(C)
X
=
R
(
C
)
such that
X
X
X
lies on the surface of the cube for some rotation
R
(
A
)
=
B
.
\mathcal R(A)=B.
R
(
A
)
=
B
.
1
1
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Inequality with cubed reciprocals
Show that the inequality
∏
k
=
2
n
(
1
−
1
k
3
)
>
1
2
\prod_{k=2}^n\left(1-\frac{1}{k^3}\right)>\frac12
k
=
2
∏
n
(
1
−
k
3
1
)
>
2
1
holds for every positive integer
n
>
1.
n>1.
n
>
1.