International Competitions 1989

Prove that 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.

In an acute-angled triangle $ABC$ the internal bisector of angle $A$ meets the circumcircle of the triangle again at $A_1$. Points $B_1$ and $C_1$ are defined similarly. Let $A_0$ be the point of intersection of the line $AA_1$ with the external bisectors of angles $B$ and $C$. Points $B_0$ and $C_0$ are defined similarly. Prove that:

(i) The area of the triangle $A_0B_0C_0$ is twice the area of the hexagon $AC_1BA_1CB_1$.

(ii) The area of the triangle $A_0B_0C_0$ is at least four times the area of the 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 any point $P$ of $S$ there are at least $k$ points of $S$ equidistant from $P$.

Prove that:

$$k < \frac{1}{2} + \sqrt{2n}.$$

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 $(x_1, x_2, \ldots, x_m)$ of the set $\{1, 2, \ldots, 2n\}$, where $n$ is a positive integer, is said to have property $P$ 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 $P$ than without.