IMO 1989

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.