1998 Kurschak Competition

Part of Kürschák Math Competition

Subcontests

(3)

1

N=? points, every triangle contains one point

For which integers

N\ge 3

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?

1

Polynomial in Z[x] - values at 1,...,n are powers of 2

Prove that for every positive integer

n

, there exists a polynomial with integer coefficients whose values at points

1,2,\dots,n

are pairwise different powers of

2

1

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

that divides every term of the sequence?