Prove that in the set $ \{1,2, \ldots, 1989\}$ can be expressed as the disjoint union of subsets $ A_i, \{i = 1,2, \ldots, 117\}$ such that i.) each $ A_i$ contains 17 elements ii.) the sum of all the elements in each $ A_i$ is the same.

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 = 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that i.) no three points of $ S$ are collinear, and ii.) for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$ Prove that: \[ k < \frac {1}{2} + \sqrt {2 \cdot n} \]

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB = AD + BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP = h + AD$ and $ BP = h + BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} + \frac {1}{\sqrt {BC}} \]

Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

A permutation $ \{x_1, x_2, \ldots, x_{2n}\}$ of the set $ \{1,2, \ldots, 2n\}$ where $ n$ is a positive integer, is said to have property $ T$ if $ |x_i - x_{i + 1}| = n$ for at least one $ i$ in $ \{1,2, \ldots, 2n - 1\}.$ Show that, for each $ n$, there are more permutations with property $ T$ than without.