Problem - 2107
There are $99$ points on a plane. Among any three points, at least two of them are not more than $1$ unit length apart. Prove: it is possible to cover $50$ of these points using a unit circle.
Let point $X$ be the one which has the most number of edges less than the unit length. Let that number be $n$. If we can prove $n \ge 49$, then we can draw a unit circle centered at $X$ which satisfy the requirement.
If $n < 49$, then there are at least $99-50=49$