I received a mailer from United in December or January offering 60,000 bonus miles, and 5,000 more for adding an authorized user. The offer expired on January 29, but I've gotten similar mailings from time to time, so watch your mail.

Does it have to be a UA credit card? The Chase Sapphire cards earn Ultimate Rewards points which can be converted to UA miles on a 1:1 basis. And, as noted, miles in other FFPs can be redeemed for travel on UA flights -- subject to partner-award availability, of course. You might, for example, check award availability for the desired UA flights on Air Canada's website. (Air Canada's Aeroplan FFP is a transfer partner for both Amex MR points and Chase UR points.)

You are probably going to need sign-up bonuses from at least two or three cards to amass all the points/miles you will need for three one-ways.