I'm an international Masters student, aiming for a certain reasonably competitive pure math PhD program in the US next year. In Dan C's answer here, he suggests to try and get a recommendation letter from one of the faculty in the school you apply to, by doing some work under him/her. There are a couple of professors in the program whose work interest me, and one of whom I might select as my advisor (given that I'm selected for the program, of course. And assuming my interests don't shift by the next year). However, since I'm not from the US, it would not be possible for me to meet the professor in person.
Given this premise, is it appropriate to contact the professor asking for a possible summer project through online correspondence? Also, should I mention that I would be applying to the program next year?