Hey man, I just wanted to add that I don't think you "Messed this up". You obviously haven't been in school for a while

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I just went through something similar with my exams, I kept up on my Home work,studies etc. but not everyone does... some people begin cramming 3 weeks in advance, this is where ALL my girls completely fell off the chart, and I was all caught up, so just felt alone... yes you want to text/call, and yes you do come off as desperate, but don't worry, after exams she'll want to celebrate I'm sure

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The only reason shes "ignoring" you is just to focus, its not personal, just really stressful for her right now,school is serious business.
If you see her at the gym yes go say hi, and joke with her. I usually ask the girls "So do all the text books start to smell the same after a while?"

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Hope all goes well champ!