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National and Regional Contests
Hungary Contests
Kürschák Math Competition
1980 Kurschak Competition
1980 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
2
1
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1/n =1/x +1/y , b has a prime divisor of form 4k-1
Let
n
>
1
n > 1
n
>
1
be an odd integer. Prove that a necessary and sufficient condition for the existence of positive integers
x
x
x
and
y
y
y
satisfying
4
n
=
1
x
+
1
y
\frac{4}{n}=\frac{1}{x}+\frac{1}{y}
n
4
=
x
1
+
y
1
is that
n
n
n
has a prime divisor of the form
4
k
−
1
4k - 1
4
k
−
1
.
3
1
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2 tennis clubs consisting of 1000 and 1001 members
In a certain country there are two tennis clubs consisting of
1000
1000
1000
and
1001
1001
1001
members respectively. All the members have different playing strength, and the descending order of palying strengths in each club is known. Find a procedure which determines, within
11
11
11
games, who is in the
1001
1001
1001
st place among the
2001
2001
2001
players in these clubs. It is assumed that a stronger player always beats a weaker one.
1
1
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points of space are coloured with 5 colours
The points of space are coloured with five colours, with all colours being used. Prove that some plane contains four points of different colours.