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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1986 Czech And Slovak Olympiad IIIA
1986 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
3
1
Hide problems
entire space can be partitioned into crosses made of 7 unit cubes
Prove that the entire space can be partitioned into “crosses” made of seven unit cubes as shown in the picture. https://cdn.artofproblemsolving.com/attachments/2/b/77c4a4309170e8303af321daceccc4010da334.png
4
1
Hide problems
A\cap l_j =\emptyset = B\cap l_j, A\cap D_j \ne \emptyset \ne B\cap D_j
Let
C
1
,
C
2
C_1,C_2
C
1
,
C
2
, and
C
3
C_3
C
3
be points inside a bounded convex planar set
M
M
M
. Rays
l
1
,
l
2
,
l
3
l_1,l_2,l_3
l
1
,
l
2
,
l
3
emanating from
C
1
,
C
2
,
C
3
C_1,C_2,C_3
C
1
,
C
2
,
C
3
respectively partition the complement of the set
M
∪
l
1
∪
l
2
∪
l
3
M \cup l_1 \cup l_2 \cup l_3
M
∪
l
1
∪
l
2
∪
l
3
into three regions
D
1
,
D
2
,
D
3
D_1,D_2,D_3
D
1
,
D
2
,
D
3
. Prove that if the convex sets
A
A
A
and
B
B
B
satisfy
A
∩
l
j
=
∅
=
B
∩
l
j
A\cap l_j =\emptyset = B\cap l_j
A
∩
l
j
=
∅
=
B
∩
l
j
and
A
∩
D
j
≠
∅
≠
B
∩
D
j
A\cap D_j \ne \emptyset \ne B\cap D_j
A
∩
D
j
=
∅
=
B
∩
D
j
for
j
=
1
,
2
,
3
j = 1,2,3
j
=
1
,
2
,
3
, then
A
∩
B
≠
∅
A\cap B \ne \emptyset
A
∩
B
=
∅
1
1
Hide problems
(B \cup C) -(B \cap C) has an even number of elements
Given
n
∈
N
n \in N
n
∈
N
, let
A
A
A
be a family of subsets of
{
1
,
2
,
.
.
.
,
n
}
\{1,2,...,n\}
{
1
,
2
,
...
,
n
}
. If for every two sets
B
,
C
∈
A
B,C \in A
B
,
C
∈
A
the set
(
B
∪
C
)
−
(
B
∩
C
)
(B \cup C) -(B \cap C)
(
B
∪
C
)
−
(
B
∩
C
)
has an even number of elements, find the largest possible number of elements of
A
A
A
.
2
1
Hide problems
integer polynomial with P(x_1) = P(x_2) = ... = P(x_m) = 1, no integer roots
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with integer coefficients of degree
n
≥
3
n \ge 3
n
≥
3
. If
x
1
,
.
.
.
,
x
m
x_1,...,x_m
x
1
,
...
,
x
m
(
n
≥
m
≥
3
n\ge m\ge3
n
≥
m
≥
3
) are different integers such that
P
(
x
1
)
=
P
(
x
2
)
=
.
.
.
=
P
(
x
m
)
=
1
P(x_1) = P(x_2) = ... = P(x_m) = 1
P
(
x
1
)
=
P
(
x
2
)
=
...
=
P
(
x
m
)
=
1
, prove that
P
P
P
cannot have integer roots$.
5
1
Hide problems
a_1 = 1, a_{n+2} = 2a_{n+1} - a_n +2 , a_na_{n+1} = a_m
A sequence of natural numbers
a
1
,
a
2
,
.
.
.
a_1,a_2,...
a
1
,
a
2
,
...
satisfies
a
1
=
1
,
a
n
+
2
=
2
a
n
+
1
−
a
n
+
2
a_1 = 1, a_{n+2} = 2a_{n+1} - a_n +2
a
1
=
1
,
a
n
+
2
=
2
a
n
+
1
−
a
n
+
2
for
n
∈
N
n \in N
n
∈
N
. Prove that for every natural
n
n
n
there exists a natural
m
m
m
such that
a
n
a
n
+
1
=
a
m
a_na_{n+1} = a_m
a
n
a
n
+
1
=
a
m
.
6
1
Hide problems
subset with any m,n satisfy |m-n| \ge mn/25
Assume that
M
⊂
N
M \subset N
M
⊂
N
has the property that every two numbers
m
,
n
m,n
m
,
n
of
M
M
M
satisfy
∣
m
−
n
∣
≥
m
n
/
25
|m-n| \ge mn/25
∣
m
−
n
∣
≥
mn
/25
. Prove that the set
M
M
M
contains no more than
9
9
9
elements. Decide whether there exists such set M.