Is it possible to develop ADHD as a teenager even if you didn't have any problems beforehand?

Possible. If you did not have symptoms of ADHD as a young child, the only way to acquire it is by having a brain injury or brain infection. There are many conditions that can mimic ADHD. Start with an evaluation by your primary care doctor, and if necessary have a neuropsychological evaluation.
Not really, but. Adhd is assumed to be a developmental disorder, but bright children with milder forms of inattentive adhd might not get diagnosed until middle school where executive function demands and organizational demands increase and workloads and accelerated learning starts to surpass their simple abilities to power through the inefficiency caused by their adhd.
Yes/no. Usually one inherits adhd, but in some situations a similar syndrome can be acquired, as in post-chemotherapy, recurrent minimal head injury, obstructive sleep apnea, among others. Adhd typically presents in 4th, 7th and 11th grades. So one could have it but not suffer its consequences until later.
Probably not. One does not "develop" adhd. Perhaps, in general, you are very intelligent. Then, as a kid, you have no noticeable problems in school. But, once things get more challenging, in the upper grades, the disorder becomes more obvious.