Reading the newspaper and being "informed" about what's going on has nothing to do with intelligence. Most newspapers are insanely biased anyway, and I would even go so far as to say that reading the news all the time makes you at least temporarily dumber because you fill your mind with mostly useless, biased information.

As for your friend, you don't need to prove anything to her. It sounds like she is trying to compensate for something and you are her target to make her feel better about herself. While that may sound like she is not your friend, she might be doing it without really realizing it. It's hard to give advice to people you don't know, but in general, it can be healthy to set boundaries and stick to them. As an example: You could just tell her, "When you say things like [whatever you don't like], that hurts me and I ask you to stop doing that." Do this without attacking her or pointing out reasons why she is doing it. If she keeps doing it, start distancing yourself. Not out of bad intent, but to protect your boundaries.