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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1973 Kurschak Competition
1973 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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3d combo geo with (2n-3)/4 tetrahedra
n
>
4
n > 4
n
>
4
planes are in general position (so every
3
3
3
planes have just one common point, and no point belongs to more than
3
3
3
planes). Show that there are at least
2
n
−
3
4
\frac{2n-3}{ 4}
4
2
n
−
3
tetrahedra among the regions formed by the planes.
2
1
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d(r) = distance of the nearest lattice point from circle (O,r)
For any positive real
r
r
r
, let
d
(
r
)
d(r)
d
(
r
)
be the distance of the nearest lattice point from the circle center the origin and radius
r
r
r
. Show that
d
(
r
)
d(r)
d
(
r
)
tends to zero as
r
r
r
tends to infinity.
1
1
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binomials in arithmetic progression
For what positive integers
n
,
k
n, k
n
,
k
(with
k
<
n
k < n
k
<
n
) are the binomial coefficients
(
n
k
−
1
)
,
(
n
k
)
,
(
n
k
+
1
)
{n \choose k- 1} \,\,\, , \,\,\, {n \choose k} \,\,\, , \,\,\, {n \choose k + 1}
(
k
−
1
n
)
,
(
k
n
)
,
(
k
+
1
n
)
three successive terms of an arithmetic progression?