How Many Flips Prove the Edge

A vendor sells you a coin-flipping machine and claims it is unbiased. You suspect it actually lands heads $55\%$ of the time. Flips are independent and you may run as many as you like.

(a) Using a quick two-sigma argument, roughly how many flips do you need before a $55\%$ machine would show up as clearly non-random? (b) Do it properly: how many flips give a one-sided $5\%$ test an $80\%$ chance of detecting the bias? (c) A colleague flipped the machine $100$ times, got $60$ heads, and wants to declare victory. What do you tell him?