Thanks for posting the above...(official time for the above was 12 second, as the timer starts when the question is flashed on the screen - although one can hear the buzzer even before the person finished asking the problem)
I know, as I have seen this for a long time, this part (countdown round) is, for the kids, most fun part.
Kids have fun even though many states don't have it or run it unofficially (that is the winner here is not the winner of the whole competition - winner is the one who wins the written part, where accuracy counts and speed is no issue, as long as you finish the whole exam)
In national, top 12 people (selected from 3 part written exam) take part in this event, and this event is open to public, and sometimes televised.
Another event, called masters round, the top 4 people (from written part), get about half an hour or so to solve a slightly tougher problem, and then present the solution (15 minutes or so) (or partial solution) to judges (and public/coaches/parents can watch) who can ask questions about method etc. It is a good place to notice young mathematicians. (This part, unfortunately is no longer there now)
Here is one question to try.. please post the answer if you wish.. (I see it may interest some of the graph theorists here - The problem is VERY simple looking, is not too hard to solve, and is doable in finite time)
Let n be the number of points on the circumference of a circle and r the maximum number of non-overlapping regions formed by connecting each of the n points, with a line segment, to every other point on the circle including its two neighboring (adjacent) points.
1. Find the corresponding values for r when n=1,2,3,4,5,6
2. Find a pattern relating n and r (use the pattern to predict value of r when n=10)
3. Find a recursive or explicit equation, or both, for r
Getting the value for small values of n is very simple. (For n=1, r=1 (there is only one region - the whole circle , n=2, r=2 (Circle cut in two parts), n=3 (r=4 - Triangle inside a circle - four regions etc))