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?

If $a,b,c$ are positive numbers with $a^2 +b^2 +c^2 = 1$, prove that $a+b+c+\frac{1}{abc} \ge 4\sqrt3$