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Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1998 Kurschak Competition
1998 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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N=? points, every triangle contains one point
For which integers
N
≥
3
N\ge 3
N
≥
3
can we find
N
N
N
points on the plane such that no three are collinear, and for any triangle formed by three vertices of the points’ convex hull, there is exactly one point within that triangle?
2
1
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Polynomial in Z[x] - values at 1,...,n are powers of 2
Prove that for every positive integer
n
n
n
, there exists a polynomial with integer coefficients whose values at points
1
,
2
,
…
,
n
1,2,\dots,n
1
,
2
,
…
,
n
are pairwise different powers of
2
2
2
.
1
1
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Gcd's of terms in an infinite sequence
Is there an infinite sequence of positive integers where no two terms are relatively prime, no term divides any other term, and there is no integer larger than
1
1
1
that divides every term of the sequence?