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National and Regional Contests
Hungary Contests
Kürschák Math Competition
2007 Kurschak Competition
2007 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Points on a plane
Prove that any finite set
H
H
H
of lattice points on the plane has a subset
K
K
K
with the following properties:[*]any vertical or horizontal line in the plane cuts
K
K
K
in at most
2
2
2
points, [*]any point of
H
∖
K
H\setminus K
H
∖
K
is contained by a segment with endpoints from
K
K
K
.
2
1
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Relative primes in an arithmetic sequence
Prove that if from any
2007
2007
2007
consecutive terms of an infinite arithmetic progression of integers starting with
2
2
2
, one can choose a term relatively prime to all the
2006
2006
2006
other terms, then there is also a term amongst any
2008
2008
2008
consecutive terms relatively prime to the rest.
1
1
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Cards around a circle
We have placed
n
>
3
n>3
n
>
3
cards around a circle, facing downwards. In one step we may perform the following operation with three consecutive cards. Calling the one on the center
B
B
B
, the two on the ends
A
A
A
and
C
C
C
, we put card
C
C
C
in the place of
A
A
A
, then move
A
A
A
and
B
B
B
to the places originally occupied by
B
B
B
and
C
C
C
, respectively. Meanwhile, we flip the cards
A
A
A
and
B
B
B
.Using a number of these steps, is it possible to move each card to its original place, but facing upwards?