im guessing he scraped a pass with rigorous studying, or perhaps he's not so bad at memorising and shit when he tries, we wouldn't know as we havent really seen him try before.
I mean character can change. With his determination, it is highly likely he did study vigorously to pass the exam. Its not far fetch.
He pulled an Elle Woods on us
He's not dumb, he just wasn't motivated. His parents are an architect and engineer, and his sister is clearly intelligent beyond her years, he has good genetics.
Honestly, it seems he was uninterested in grades or putting any effort towards it. Remind me of my cousin, who is a wiz at math and science but doesn't put the effort in, used to drive his teachers and parents mad.
Wait, isn't Tsugumi passing the entrance exam inconsistent with the idea that Tsugumi is kind of dumb?