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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1985 Czech And Slovak Olympiad IIIA
1985 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
Hide problems
a_{k+1} divides a_1+a_2+...+a_k
Prove that for every natural number
n
>
1
n > 1
n
>
1
there exists a suquence
a
1
a_1
a
1
,
a
2
a_2
a
2
,
.
.
.
...
...
,
a
n
a_n
a
n
of the numbers
1
,
2
,
.
.
.
,
n
1,2,...,n
1
,
2
,
...
,
n
such that for each
k
∈
{
1
,
2
,
.
.
.
,
n
−
1
}
k \in \{1,2,...,n-1\}
k
∈
{
1
,
2
,
...
,
n
−
1
}
the number
a
k
+
1
a_{k+1}
a
k
+
1
divides
a
1
+
a
2
+
.
.
.
+
a
k
a_1+a_2+...+a_k
a
1
+
a
2
+
...
+
a
k
.
5
1
Hide problems
numbere in a triangular table
A triangular table with
n
n
n
rows and
n
n
n
columns is given, the
i
i
i
-th row ends with a field in the
v
v
v
-th column. In each field of the table, some of the numbers
1
,
2
,
.
.
.
,
n
1,2,..., n
1
,
2
,
...
,
n
are written such that for each
k
∈
1
,
2
,
.
.
.
,
n
k \in {1, 2,..., n}
k
∈
1
,
2
,
...
,
n
all the numbers
1
,
2
,
.
.
.
,
n
1,2,..., n
1
,
2
,
...
,
n
occur in the union of the
k
k
k
-th row and the
k
k
k
-th column. Prove that for odd
n
n
n
, each of the numbers
1
,
2
,
.
.
.
,
n
1,2,..., n
1
,
2
,
...
,
n
is written in the last box of a row. https://cdn.artofproblemsolving.com/attachments/f/9/2aed55628edb1505c7de27c152127b04d8d991.png
4
1
Hide problems
unusual locus, 2 fixed lines and a fixed point
Two straight lines
p
,
q
p, q
p
,
q
are given in the plane and on the straight line
q
q
q
there is a point
F
F
F
,
F
∉
p
F \not\in p
F
∈
p
. Determine the set of all points
X
X
X
that can be obtained by this construction: In the plane we choose a point
S
S
S
that lies neither on
p
p
p
nor on
q
q
q
, and we construct a circle
k
k
k
with center
S
S
S
that is tangent to the line
p
p
p
. On the circle
k
k
k
we choose a point
T
T
T
such that so that
S
T
∥
q
ST \parallel q
ST
∥
q
. If the line
F
T
FT
FT
intersects the line
p
p
p
at the point
U
U
U
,
X
X
X
is the intersection of the lines
S
U
SU
S
U
and
q
q
q
3
1
Hide problems
exist vectors with sum <= \sqrt2/8, among n vectors with sum <=1
If
u
1
→
,
u
2
→
,
.
.
.
,
u
n
→
\overrightarrow{u_1},\overrightarrow{u_2}, ...,\overrightarrow{u_n}
u
1
,
u
2
,
...
,
u
n
be vectors in the plane such that the sum of their lengths is at least
1
1
1
, then between them we find vectors whose sum is a vector of length at least
2
/
8
\sqrt2/8
2
/8
. Prove it.
2
1
Hide problems
{i, j, k} = {1, 2, 3}, (x \in A_i, y\in A_j) > (x + y \in A_k, x - y \in A_k).
Let
A
1
,
A
2
,
A
3
A_1, A_2, A_3
A
1
,
A
2
,
A
3
be nonempty sets of integers such that for
{
i
,
j
,
k
}
=
{
1
,
2
,
3
}
\{i, j, k\} = \{1, 2, 3\}
{
i
,
j
,
k
}
=
{
1
,
2
,
3
}
holds
(
x
∈
A
i
,
y
∈
A
j
)
⇒
(
x
+
y
∈
A
k
,
x
−
y
∈
A
k
)
.
(x \in A_i, y\in A_j) \Rightarrow (x + y \in A_k, x - y \in A_k).
(
x
∈
A
i
,
y
∈
A
j
)
⇒
(
x
+
y
∈
A
k
,
x
−
y
∈
A
k
)
.
Prove that at least two of the sets
A
1
,
A
2
,
A
3
A_1, A_2, A_3
A
1
,
A
2
,
A
3
are equal. Can any of these sets be disjoint?
1
1
Hide problems
line cutting a regular 1985-gon
A regular
1985
1985
1985
-gon is given in the plane. Let's pass a straight line through each side of it. Determine the number of parts into which these lines divide the plane.