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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1982 Czech and Slovak Olympiad III A
1982 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
5
1
Hide problems
comparing terms of seqeunce
Given is a sequence of real numbers
{
a
n
}
n
=
1
∞
\{a_n\}^{\infty}_{n=1}
{
a
n
}
n
=
1
∞
such that
a
n
≠
a
m
a_n \ne a_m
a
n
=
a
m
for
n
≠
m
,
n\ne m,
n
=
m
,
given is a natural number
k
k
k
. Construct a simple representation
P
P
P
of the set
1
,
2
,
.
.
.
,
20
k
1,2,...,20k
1
,
2
,
...
,
20
k
into the set of natural numbers to apply
a
p
(
1
)
<
a
p
(
2
)
<
.
.
.
<
a
p
(
10
)
a_{p(1)}<a_{p(2)}<...<a_{p(10)}
a
p
(
1
)
<
a
p
(
2
)
<
...
<
a
p
(
10
)
a
p
(
10
)
>
a
p
(
11
)
>
.
.
.
>
a
p
(
20
)
a_{p(10)}>a_{p(11)}>...>a_{p(20)}
a
p
(
10
)
>
a
p
(
11
)
>
...
>
a
p
(
20
)
a
p
(
20
)
<
a
p
(
21
)
<
.
.
.
<
a
p
(
30
)
a_{p(20)}<a_{p(21)}<...<a_{p(30)}
a
p
(
20
)
<
a
p
(
21
)
<
...
<
a
p
(
30
)
.
.
.
...
...
a
p
(
20
k
−
10
)
>
a
p
(
20
k
−
9
)
>
.
.
.
>
a
p
(
20
k
)
a_{p(20k-10)}>a_{p(20k-9)}>...>a_{p(20k)}
a
p
(
20
k
−
10
)
>
a
p
(
20
k
−
9
)
>
...
>
a
p
(
20
k
)
a
p
(
10
)
>
a
p
(
30
)
>
.
.
.
>
a
p
(
(
20
k
−
10
)
)
a_{p(10)}>a_{p(30)}>...>a_{p((20k-10))}
a
p
(
10
)
>
a
p
(
30
)
>
...
>
a
p
((
20
k
−
10
))
a
p
(
1
)
<
a
p
(
20
)
<
.
.
.
<
a
p
(
20
k
)
,
a_{p(1)}<a_{p(20)}<...<a_{p(20k)},
a
p
(
1
)
<
a
p
(
20
)
<
...
<
a
p
(
20
k
)
,
[hide=original wording of first sentence]Daná je postupnos’ reálných čísel {a_} také že an \ne a_m ctm pre n\ne m, dané je prirodzené číslo k. Zostrojte prosté zobrazenie P množiny 1,2,...,20k do -množiny prirodzených . čísel , aby platilo
6
1
Hide problems
x_1^k+x_2^k+...+x_n^k=1, (1+x_1)(1+x_2)...(1+x_n)=2
Let
n
,
k
n,k
n
,
k
be given natural numbers. Determine all ordered n-tuples of non-negative real numbers
(
x
1
,
x
2
,
.
.
.
,
x
n
)
(x_1,x_2,...,x_n)
(
x
1
,
x
2
,
...
,
x
n
)
that satisfy the system of equations
x
1
k
+
x
2
k
+
.
.
.
+
x
n
k
=
1
x_1^k+x_2^k+...+x_n^k=1
x
1
k
+
x
2
k
+
...
+
x
n
k
=
1
(
1
+
x
1
)
(
1
+
x
2
)
.
.
.
(
1
+
x
n
)
=
2
(1+x_1)(1+x_2)...(1+x_n)=2
(
1
+
x
1
)
(
1
+
x
2
)
...
(
1
+
x
n
)
=
2
4
1
Hide problems
10 from 64 points in unit circle, lies on circle of radius 1/2
In a circle with a radius of
1
1
1
,
64
64
64
mutually different points are selected. Prove that
10
10
10
mutually different points can be selected from them, which lie in a circle with a radius
1
2
\frac12
2
1
.
3
1
Hide problems
convex set M with infinite lattice points
In the plane with coordinates
x
,
y
x,y
x
,
y
, find an example of a convex set
M
M
M
that contains infinitely many lattice points (i.e. points with integer coordinates), but at the same time only finitely many lattice points from
M
M
M
lie on each line in that plane.
2
1
Hide problems
| x_1x_4-x_1x_5 +x_2x_5 -x_2x_6+x_3x_6-x_3x_4| >= 4M^2
Given real numbers
x
1
x_1
x
1
,
x
2
x_2
x
2
,
x
3
x_3
x
3
,
x
4
x_4
x
4
,
x
5
x_5
x
5
,
x
6
x_6
x
6
. Let
M
M
M
denote the maximum of their absolute values. Prove that it is valid
∣
x
1
x
4
−
x
1
x
5
+
x
2
x
5
−
x
2
x
6
+
x
3
x
6
−
x
3
x
4
∣
≤
4
M
2
| x_1x_4-x_1x_5 +x_2x_5 -x_2x_6+x_3x_6-x_3x_4| \le 4M^2
∣
x
1
x
4
−
x
1
x
5
+
x
2
x
5
−
x
2
x
6
+
x
3
x
6
−
x
3
x
4
∣
≤
4
M
2
1
1
Hide problems
collinear centroids of tetrahedra
Given a tetrahedron
A
B
C
D
ABCD
A
BC
D
and inside the tetrahedron points
K
,
L
,
M
,
N
K, L, M, N
K
,
L
,
M
,
N
that do not lie on a plane. Denote also the centroids of
P
P
P
,
Q
Q
Q
,
R
R
R
,
S
S
S
of the tetrahedrons
K
B
C
D
KBCD
K
BC
D
,
A
L
C
D
ALCD
A
L
C
D
,
A
B
M
D
ABMD
A
BM
D
,
A
B
C
N
ABCN
A
BCN
do not lie on a plane. Let
T
T
T
be the centroid of the tetrahedron ABCD,
T
o
T_o
T
o
be the centroid of the tetrahedron
P
Q
R
S
PQRS
PQRS
and
T
1
T_1
T
1
be the centroid of the tetrahedron
K
L
M
N
KLMN
K
L
MN
.a) Prove that the points
T
,
T
0
,
T
1
T, T_0, T_1
T
,
T
0
,
T
1
lie in one straight line. b) Determine the ratio
∣
T
0
T
∣
:
∣
T
0
T
1
∣
|T_0T| : |T_0 T_1|
∣
T
0
T
∣
:
∣
T
0
T
1
∣
.