MK-MO 1994

Let $ABC$ be a triangle whose vertices have integer coordinates and inside of which there is exactly one point $O$ with integer coordinates. Let $D$ be the intersection of the lines $BC$ and $AO.$ Find the largest possible value of $\frac {\overline{AO}} {\overline{OD}}$ .

a) Let $x_1, x_2, ..., x_n$ ( $n> 2$ ) be negative real numbers and $x_1 + x_2 + ... + x_n = m.$ Determine the maximum value of the sum $S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n.$ b) Let $x_1, x_2, ..., x_n$ ( $n> 2$ ) be nonnegative natural numbers and $x_1 + x_2 + ... + x_n = m.$ Determine the maximum value of the sum $S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n.$

$1994$ points from the plane are given so that any $100$ of them can be selected $98$ that can be rounded (some points may be at the border of the circle) with a diameter of $1$ . Determine the smallest number of circles with radius $1$ , sufficient to cover all $1994$