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Problems
Contests
International Contests
Czech-Polish-Slovak Match
2006 Czech-Polish-Slovak Match
2006 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
6
1
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Pentagon whose diagonal intersections are collinear
Find out if there is a convex pentagon
A
1
A
2
A
3
A
4
A
5
A_1A_2A_3A_4A_5
A
1
A
2
A
3
A
4
A
5
such that, for each
i
=
1
,
…
,
5
i = 1, \dots , 5
i
=
1
,
…
,
5
, the lines
A
i
A
i
+
3
A_iA_{i+3}
A
i
A
i
+
3
and
A
i
+
1
A
i
+
2
A_{i+1}A_{i+2}
A
i
+
1
A
i
+
2
intersect at a point
B
i
B_i
B
i
and the points
B
1
,
B
2
,
B
3
,
B
4
,
B
5
B_1,B_2,B_3,B_4,B_5
B
1
,
B
2
,
B
3
,
B
4
,
B
5
are collinear. (Here
A
i
+
5
=
A
i
A_{i+5} = A_i
A
i
+
5
=
A
i
.)
5
1
Hide problems
A sequence that remains integral
Find the number of sequences
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^\infty
(
a
n
)
n
=
1
∞
of integers satisfying
a
n
≠
−
1
a_n \ne -1
a
n
=
−
1
and
a
n
+
2
=
a
n
+
2006
a
n
+
1
+
1
a_{n+2} =\frac{a_n + 2006}{a_{n+1} + 1}
a
n
+
2
=
a
n
+
1
+
1
a
n
+
2006
for each
n
∈
N
n \in \mathbb{N}
n
∈
N
.
4
1
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There exists a power of 2 with exactly k consecutive zeroes
Show that for every integer
k
≥
1
k \ge 1
k
≥
1
there is a positive integer
n
n
n
such that the decimal representation of
2
n
2^n
2
n
contains a block of exactly
k
k
k
zeros, i.e.
2
n
=
…
a
00
…
0
b
⋯
2^n = \dots a00 \dots 0b \cdots
2
n
=
…
a
00
…
0
b
⋯
with
k
k
k
zeros and
a
,
b
≠
0
a, b \ne 0
a
,
b
=
0
.
3
1
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Four real numbers with specified sum and sum of squares
The sum of four real numbers is
9
9
9
and the sum of their squares is
21
21
21
. Prove that these numbers can be denoted by
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
so that
a
b
−
c
d
≥
2
ab-cd \ge 2
ab
−
c
d
≥
2
holds.
2
1
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Distributing candies so that each child gets one
There are
n
n
n
children around a round table. Erika is the oldest among them and she has
n
n
n
candies, while no other child has any candy. Erika decided to distribute the candies according to the following rules. In every round, she chooses a child with at least two candies and the chosen child sends a candy to each of his/her two neighbors. (So in the first round Erika must choose herself). For which
n
≥
3
n \ge 3
n
≥
3
is it possible to end the distribution after a finite number of rounds with every child having exactly one candy?
1
1
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Cyclic Points form Right triangle
Five distinct points
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
and
E
E
E
lie in this order on a circle of radius
r
r
r
and satisfy
A
C
=
B
D
=
C
E
=
r
AC = BD = CE = r
A
C
=
B
D
=
CE
=
r
. Prove that the orthocentres of the triangles
A
C
D
,
B
C
D
ACD, BCD
A
C
D
,
BC
D
and
B
C
E
BCE
BCE
are the vertices of a right-angled triangle.