MK-MO 1999

In a set of $21$ real numbers, the sum of any $10$ numbers is less than the sum of the remaining $11$ numbers. Prove that all the numbers are positive.

We are given $13$ apparently equal balls, all but one having the same weight (the remaining one has a different weight). Is it posible to determine the ball with the different weight in $3$ weighings?

Let the two tangents from a point $A$ outside a circle $k$ touch $k$ at $M$ and $N$ . A line $p$ through $A$ intersects $k$ at $B$ and $C$ , and $D$ is the midpoint of $MN$ . Prove that $MN$ bisects the angle $BDC$

Do there exist $100$ straight lines on a plane such that they intersect each other in exactly $1999$ points?