KGS math club/solution 11 22

The first question has been solved (you can find various papers on the net, i.e. http://2012.cccg.ca/papers/paper13.pdf), so response is YES and constitute Inaba's theorem. The proof for 10 is the less difficult one, and can be proven by probabilistic reasoning (you can try by yourself). For lower bounds >= 11 it becomes harder and you can consult the link it gave which also reference some former work on the subject.