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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1989 Czech And Slovak Olympiad IIIA
1989 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
3
1
Hide problems
[n p/q], [cn + d],
For given coprime numbers
p
>
q
>
0
p > q > 0
p
>
q
>
0
, find all pairs of real numbers
c
,
d
c,d
c
,
d
such that for the sets
A
=
{
[
n
p
q
]
,
n
∈
N
}
a
n
d
B
=
{
[
c
n
+
d
]
,
n
∈
N
}
A = \left\{ \left[n\frac{p}{q}\right] , n \in N \right\} \ \ and \ \ B = \{[cn + d], n \in N\}
A
=
{
[
n
q
p
]
,
n
∈
N
}
an
d
B
=
{[
c
n
+
d
]
,
n
∈
N
}
where
A
∩
B
=
∅
A \cap B = \emptyset
A
∩
B
=
∅
,
A
∪
B
=
N
A \cup B = N
A
∪
B
=
N
, where
N
=
{
1
,
2
,
3
,
.
.
.
}
N = \{1, 2, 3, ...\}
N
=
{
1
,
2
,
3
,
...
}
is the set of all natural numbers.
6
1
Hide problems
a_i = natural at most n, a_p = a_r \ne a_q = a_s
Consider a finite sequence
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
whose terms are natural numbers at most equal to
n
n
n
. Determine the maximum number of terms of such a sequence, if you know that every two of its neighboring terms are different and at the same time there is no quartet of terms in it such that
a
p
=
a
r
≠
a
q
=
a
s
a_p = a_r \ne a_q = a_s
a
p
=
a
r
=
a
q
=
a
s
for
p
<
q
<
r
<
s
p < q < r < s
p
<
q
<
r
<
s
.
5
1
Hide problems
P_n= number of colorings of 2xn strips ...
Consider a rectangular table
2
×
n
.
2 \times n.
2
×
n
.
Let every cell be dyed either by black or white color in a way that no
2
×
2
2\times 2
2
×
2
square is completely black. Denote
P
n
P_n
P
n
the number of such colorings. Prove that the number
P
1989
P_{1989}
P
1989
is divisible by three and find the greatest power of three that divides them.
4
1
Hide problems
triangles with an equal angle and perpendicular sides are similar
The lengths of the sides of triangle
T
′
T'
T
′
are equal to the lengths of the medians of triangle
T
T
T
. If triangles
T
T
T
and
T
′
T'
T
′
coincide in one angle, they are similar. Prove it.
2
1
Hide problems
mn line segments that connect n given points
There are
m
n
mn
mn
line segments in a plane that connect
n
n
n
given points. Prove that a sequence
V
0
V_0
V
0
,
V
1
V_1
V
1
,
.
.
.
...
...
,
V
m
V_m
V
m
of different points can be selected from them such that
V
i
−
1
V_{i-1}
V
i
−
1
and
V
i
V_i
V
i
are connected by a line (
1
≤
i
≤
m
1 \le i \le m
1
≤
i
≤
m
).
1
1
Hide problems
concyclic , AS _|_ p
Three different points
A
,
B
,
C
A, B, C
A
,
B
,
C
lying on a circle with center
S
S
S
and a line
p
p
p
perpendicular to
A
S
AS
A
S
are given in the plane. Let's mark the intersections of the line
p
p
p
with the lines
A
B
AB
A
B
,
A
C
AC
A
C
as
D
D
D
and
E
E
E
. Prove that the points
B
,
C
,
D
,
E
B, C, D, E
B
,
C
,
D
,
E
lie on the same circle.