National Competition

If the problem could be solved with $2016$ jumps, the total distance covered by these jumps would be strictly between $2\cdot 2016$ and $3\cdot 2016$ which makes a return to the original point impossible. Therefore, at least $2017$ jumps are required. This is indeed possible, for example with the following sequence of points on the circle. $$0,3,6,\dots ,2013,2015,2,5,\dots ,2012,2014,1,4,\dots ,2011,2013,0.$$