KURSCHAK 1973

For what positive integers $n, k$ (with $k < n$ ) are the binomial coefficients $${n \choose k- 1} \,\, , \,\, {n \choose k} \,\, , \,\, {n \choose k + 1}$$ three successive terms of an arithmetic progression?

For any positive real $r$ , let $d(r)$ be the distance of the nearest lattice point from the circle center the origin and radius $r$ . Show that $d(r)$ tends to zero as $r$ tends to infinity.

$n > 4$ planes are in general position (so every $3$ planes have just one common point, and no point belongs to more than $3$ planes). Show that there are at least $\frac{2n-3}{ 4}$ tetrahedra among the regions formed by the planes.