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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1993 Czech And Slovak Olympiad IIIA
1993 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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exists tetrahedron which can be partitioned into 8 congruent tetrahedra
Show that there exists a tetrahedron which can be partitioned into eight congruent tetrahedra, each of which is similar to the original one.
5
1
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f(x)+ f(y) = f(x+2xy)+ f(y-2xy), f(-1) = f(1)
Find all functions
f
:
Z
→
Z
f : Z \to Z
f
:
Z
→
Z
such that
f
(
−
1
)
=
f
(
1
)
f(-1) = f(1)
f
(
−
1
)
=
f
(
1
)
and
f
(
x
)
+
f
(
y
)
=
f
(
x
+
2
x
y
)
+
f
(
y
−
2
x
y
)
f(x)+ f(y) = f(x+2xy)+ f(y-2xy)
f
(
x
)
+
f
(
y
)
=
f
(
x
+
2
x
y
)
+
f
(
y
−
2
x
y
)
for all
x
,
y
∈
Z
x,y \in Z
x
,
y
∈
Z
4
1
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a_{n+1} equals the sum of tenth powers of decimal digits of a_n
The sequence (
a
n
a_n
a
n
) of natural numbers is defined by
a
1
=
2
a_1 = 2
a
1
=
2
and
a
n
+
1
a_{n+1}
a
n
+
1
equals the sum of tenth powers of the decimal digits of
a
n
a_n
a
n
for all
n
≥
1
n \ge 1
n
≥
1
. Are there numbers which appear twice in the sequence (
a
n
a_n
a
n
)?
3
1
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lines intersect at the circumcenter of the equilateral trapezoid
Let
A
K
L
AKL
A
K
L
be a triangle such that
∠
A
L
K
>
9
0
o
+
∠
L
A
K
\angle ALK > 90^o +\angle LAK
∠
A
L
K
>
9
0
o
+
∠
L
A
K
. Construct an isosceles trapezoid
A
B
C
D
ABCD
A
BC
D
with
A
B
∥
C
D
AB \parallel CD
A
B
∥
C
D
such that
K
K
K
lies on the side
B
C
,
L
BC, L
BC
,
L
on the diagonal
A
C
AC
A
C
and the lines
A
K
AK
A
K
and
B
L
BL
B
L
intersect at the circumcenter of the trapezoid.
2
1
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integers in fields of a 19 x 19 table
In fields of a
19
×
19
19 \times 19
19
×
19
table are written integers so that any two lying on neighboring fields differ at most by
2
2
2
(two fields are neighboring if they share a side). Find the greatest possible number of mutually different integers in such a table.
1
1
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7^n -1 is divisible by 6^n -1
Find all natural numbers
n
n
n
for which
7
n
−
1
7^n -1
7
n
−
1
is divisible by
6
n
−
1
6^n -1
6
n
−
1