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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1987 Kurschak Competition
1987 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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3n+1 members in a club
Any two members of a club with
3
n
+
1
3n+1
3
n
+
1
people plays ping-pong, tennis or chess with each other. Everyone has exactly
n
n
n
partners who plays ping-pong,
n
n
n
who play tennis and
n
n
n
who play chess. Prove that we can choose three members of the club who play three different games amongst each other.
2
1
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Old classic - points in space
Is there a set of points in space whose intersection with any plane is a finite but nonempty set of points?
1
1
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a+b=cd, c+d=ab
Find all quadruples of positive integers
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
such that
a
+
b
=
c
d
a+b=cd
a
+
b
=
c
d
and
c
+
d
=
a
b
c+d=ab
c
+
d
=
ab
.