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Problems
Contests
International Contests
Czech-Polish-Slovak Match
1996 Czech and Slovak Match
1996 Czech and Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
3
1
Hide problems
inequality with surface areas of 5 pyramids
The base of a regular quadrilateral pyramid
π
\pi
π
is a square with side length
2
a
2a
2
a
and its lateral edge has length a
17
\sqrt{17}
17
. Let
M
M
M
be a point inside the pyramid. Consider the five pyramids which are similar to
π
\pi
π
, whose top vertex is at
M
M
M
and whose bases lie in the planes of the faces of
π
\pi
π
. Show that the sum of the surface areas of these five pyramids is greater or equal to one fifth the surface of
π
\pi
π
, and find for which
M
M
M
equality holds.
4
1
Hide problems
f (x)+kx = m with at least one integral solution x for any m
Decide whether there exists a function
f
:
Z
→
Z
f : Z \rightarrow Z
f
:
Z
→
Z
such that for each
k
=
0
,
1
,
.
.
.
,
1996
k =0,1, ...,1996
k
=
0
,
1
,
...
,
1996
and for any integer
m
m
m
the equation
f
(
x
)
+
k
x
=
m
f (x)+kx = m
f
(
x
)
+
k
x
=
m
has at least one integral solution
x
x
x
.
5
1
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two sets of intervals on a line
Two sets of intervals
A
,
B
A ,B
A
,
B
on the line are given. The set
A
A
A
contains
2
m
−
1
2m-1
2
m
−
1
intervals, every two of which have an interior point in common. Moreover, every interval from
A
A
A
contains at least two disjoint intervals from
B
B
B
. Show that there exists an interval in
B
B
B
which belongs to at least
m
m
m
intervals from
A
A
A
.
2
1
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binary operation, (a ⋆ b) ⋆ b= a , a ⋆ (a ⋆ b)= b
Let ⋆ be a binary operation on a nonempty set
M
M
M
. That is, every pair
(
a
,
b
)
∈
M
(a,b) \in M
(
a
,
b
)
∈
M
is assigned an element
a
a
a
⋆
b
b
b
in
M
M
M
. Suppose that ⋆ has the additional property that
(
a
(a
(
a
⋆
b
)
b)
b
)
⋆
b
=
a
b= a
b
=
a
and
a
a
a
⋆
(
a
(a
(
a
⋆
b
)
=
b
b)= b
b
)
=
b
for all
a
,
b
∈
M
a,b \in M
a
,
b
∈
M
. (a) Show that
a
a
a
⋆
b
=
b
b = b
b
=
b
⋆
a
a
a
for all
a
,
b
∈
M
a,b \in M
a
,
b
∈
M
. (b) On which finite sets
M
M
M
does such a binary operation exist?
1
1
Hide problems
p prime iff 1 of a+b-6ab+(p-1)/6 , a+b+6ab+(p-1)/6 \in Z
Show that an integer
p
>
3
p > 3
p
>
3
is a prime if and only if for every two nonzero integers
a
,
b
a,b
a
,
b
exactly one of the numbers
N
1
=
a
+
b
−
6
a
b
+
p
−
1
6
N_1 = a+b-6ab+\frac{p-1}{6}
N
1
=
a
+
b
−
6
ab
+
6
p
−
1
,
N
2
=
a
+
b
+
6
a
b
+
p
−
1
6
N_2 = a+b+6ab+\frac{p-1}{6}
N
2
=
a
+
b
+
6
ab
+
6
p
−
1
is a nonzero integer.
6
1
Hide problems
Nice Conditions!
The points
E
E
E
and
D
D
D
lie in the interior of sides
A
C
AC
A
C
and
B
C
BC
BC
, respectively, of a triangle
A
B
C
ABC
A
BC
. Let
F
F
F
be the intersection of the lines
A
D
AD
A
D
and
B
E
BE
BE
.Show that the area of the traingles
A
B
C
ABC
A
BC
and
A
B
F
ABF
A
BF
satisfies:
S
A
B
C
S
A
B
F
=
∣
A
C
∣
∣
A
E
∣
+
∣
B
C
∣
∣
B
D
∣
−
1
\frac{S_{ABC}}{S_{ABF}} = \frac{\mid{AC}\mid}{\mid{AE} \mid} + \frac{\mid{BC}\mid}{\mid{BD}\mid} - 1
S
A
BF
S
A
BC
=
∣
A
E
∣
∣
A
C
∣
+
∣
B
D
∣
∣
BC
∣
−
1
.