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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1996 Kurschak Competition
1996 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Broken line from chosen diagonals of n-gon
Let
n
n
n
and
k
k
k
be arbitrary non-negative integers. Suppose we have drawn
2
k
n
+
1
2kn+1
2
kn
+
1
(different) diagonals of a convex
n
n
n
-gon. Show that there exists a broken line formed by
2
k
+
1
2k+1
2
k
+
1
of these diagonals that passes through no point more than once. Prove also that this is not necessarily true when we draw only
k
n
kn
kn
diagonals.
2
1
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Two-country conference
Two countries (
A
A
A
and
B
B
B
) organize a conference, and they send an equal number of participants. Some of them have known each other from a previous conference. Prove that one can choose a nonempty subset
C
C
C
of the participants from
A
A
A
such that one of the following holds: [*]the participants from
B
B
B
each know an even number of people in
C
C
C
, [*]the participants from
B
B
B
each know an odd number of participants in
C
C
C
.
1
1
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Inequality - trapezoid with perp. diagonals
Prove that in a trapezoid with perpendicular diagonals, the product of the legs is at least as much as the product of the bases.