5. In a quiz, there are three doors. Behind one of the doors is a prize. You may ask the quizmaster if the prize is behind the left door. You may also ask if the prize is behind the right door. You can ask each of these two questions multiple times in an order that you can choose. Each time, the quizmaster answers with 'yes' or 'no'. In the entire series of questions, he may lie at most 10 times. You must specify in advance how many questions you will ask (but which questions they are can depend on the quizmaster's answers).

What is the smallest number you can name to still be sure that you will know where the prize is after the series of questions?
5. The smallest number of questions needed is 32. First, we will show a strategy to find the location of the prize in 32 questions.

At the beginning, you keep asking the quizmaster if the prize is behind the left door until you are certain of the answer. You only know the answer with 100% certainty after receiving the same answer 11 times. The quizmaster can lie at most 10 times. Suppose the quizmaster has lied $n$ times at that point. Then you have asked a total of $11+n$ questions, and the quizmaster can still lie $10-n$ times.

Next, you keep asking the quizmaster if the prize is behind the right door until you are 100% certain of the answer. The quizmaster can still lie $10-n$ times, so at most $2(10-n)+1=20-2n+1$ questions are needed. In total, you have then asked $32-n$ questions. A strategy with 32 questions is therefore possible.

With only 31 questions, it is not always possible to determine the location of the prize. We will now show how the quizmaster can ensure this. At the beginning, as long as you have asked no more than ten questions about each door, he answers 'no' every time. For simplicity, we can assume that the first door you ask eleven questions about is the left door. We now consider the situation where the prize is not behind the left door (which can happen). In this case, we show that the quizmaster can ensure that after 31 questions, you do not know whether the prize is behind the middle door or the right door.

The quizmaster has so far always told the truth about the left door and will continue to do so: he always says 'no' about the left door. About the right door, the quizmaster continues to say 'no' until the tenth question about this door. After that, he answers the next ten questions about this door with 'yes'. Since you ask at most 20 questions about the right door in total (because you have already asked at least 11 questions about the left door), the quizmaster does not have to lie more than 10 times. Whether the prize is behind the middle door or the right door, the quizmaster gives the same 31 answers in both cases. Therefore, it is impossible to determine behind which of the two doors the prize is located.