Club of Abusers

To enter the Club of Abusers, you have to pass a qualifying exam of 100 True/False questions; the passing grade is 51 questions correct. The questions are so obscure, and rumors say that the graders also mark them in such a way to try to prevent you from passing.

However, for some reason, you are allowed to retake the exam as many times as you wish. As long as any one of the exams receive a passing grade, you are admitted into the club.

To prove that you're worthy to enter the club, you need to abuse this qualifying exam. Take the exam enough times so that at least one is guaranteed to have a passing grade, no matter what the actual answers are. Of course, don't take it too much times; you don't want to spend too long just to enter the club.

To recap:

• You have \(k\) exam sheets. For each exam, you have 100 questions; all the exams are the same, including the question order (and thus have the same sequence of correct answers).
• For each question, give an answer of either True or False (each question has two choices).
• You must submit all exam sheets at the same time; there is no feedback for any exam sheet that you can use to fill in another exam sheet.
• If any of the exams is correct on at least 51 questions, you win.
• Find the smallest positive integer \(k\) that allows you to always win with an appropriate strategy.