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National and Regional Contests
Hungary Contests
Kürschák Math Competition
1954 Kurschak Competition
1954 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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tournament is arranged amongst a finite number of people
A tournament is arranged amongst a finite number of people. Every person plays every other person just once and each game results in a win to one of the players (there are no draws). Show that there must a person
X
X
X
such that, given any other person
Y
Y
Y
in the tournament, either
X
X
X
beat
Y
Y
Y
, or
X
X
X
beat
Z
Z
Z
and
Z
Z
Z
beat
Y
Y
Y
for some
Z
Z
Z
.
2
1
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every planar section of a three-dimensional body B is a disk
Every planar section of a three-dimensional body
B
B
B
is a disk. Show that B must be a ball.
1
1
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AB < AC if ABCD convex with AB + BD = AC + CD
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral with
A
B
+
B
D
=
A
C
+
C
D
AB + BD = AC + CD
A
B
+
B
D
=
A
C
+
C
D
. Prove that
A
B
<
A
C
AB < AC
A
B
<
A
C
.