Wow, I cannot believe how many people can't discuss this in a civil manner without resorting to personal attacks and wide generalizations.
Ok, so lets define the term sexism first, from Dictionary.com;
Sexism: attitudes or behavior based on traditional stereotypes of sexual roles.
What does this mean, well, if you do something solely because of your sex or another person's sex, that's sexism. Lets take a closer look;
Is it sexism?
1.) Opening a door solely because there's a woman coming?
A. Sexist
2.) Opening a door solely because there's a PERSON coming?
A. Not sexist
Both of these can be classified as "gentlemanly" acts, however, one is clearly sexist-based, the other is not. However, it begs the question of why call it "gentlemanly" when really, its just about respect, kindness, and charity. What do you call a woman that behaves in the same manner? A gentlewoman? The title, really, is not doing anyone any favours as it stereotypes the role of a male (ie. a good man should act like a gentleman when instead, all should strive to act kindly and generously, regardless of sex).
Inversely, the same can be said for a woman, would they have an equivalent title. For example;
Is it sexism?
1.) A woman cooks a meal solely because there's a man coming?
A. Sexist
2.) A woman cooks a meal solely because there's a PERSON coming?
A. Not sexist
In all of my examples, its a kind act that's being done, but its the motivating factor behind the act that establishes whether its sexist or not. Otherwise, its just a kind act.
PS. For those of you making general observations about "I do something for a girl (really a person) I like, is that sexist?". No and yes. Sexual orientation is generally sexist (ie read bisexualism), that's why its referred to as your "Sex". But ultimately, your motivating factor is that you're attracted to the person and are trying to gain their favor, that is the real motivation behind your action(s). Their gender is only what fuels your attraction to them. After all, if you didn't find them to be an "attractive" female (again, person really), your behaviour would not be inclined to be try and gain their interest.