A friend who dosen't open up

I suspect she's not opening up to you because so far it hasn't gotten her anywhere.

Why should she tell you her problems if you say nothing about them when she does?

So here's an idea. Don't fish for problems. Instead, ask, "How are you ... really?"

Not everyone will answer that question, or answer it at once. Be patient. And if/when she does answer you (and she may never; you'll need to learn to accept that if that's what happens), listen.

As for offering solutions, not everyone wants them, even if they open up all the time. Sometimes, people just want to vent and be heard. So ask -- "Do you want me to try to help you fix things, or something else?" Follow her lead in this area.

BTW, your solutions might not be as bad/wrongheaded as you fear. But ask first, if she wants solutions, or not.

And recognize, too, that she may not feel your friendship is as close as you believe it to be.

Seconding these two important points.

Just a simple possible observation, but are you willing to hear out and help her with her problems... because ... you want to strengthen the friendship... or you want to get out of the friendzone and become more intimate with her in pushing the friendship beyond its current status... a less platonic status of boyfriend/girlfriend or girlfriend/girlfriend?

She may suspect this is the case and chooses to not provide any further encouragement towards this outcome.

Just a mere hypothetical question, not an accusation.