5. In an intelligence competition show, there are three doors, one of which hides a prize. The contestant can ask the host if the prize is behind the left (or right) door. The contestant can ask the host multiple times in any order of the chosen doors, and the host will answer "yes" or "no" each time. It is known that before the official competition, the contestant needs to inform the host in advance of the number of times they will choose a door, and the host can lie at most 10 times. How many times does the contestant need to ask at least to definitely determine the door hiding the prize?
5. Ask at least 32 times.

First, ask the host if the prize is behind the left door. Since the host can lie at most 10 times, repeat this step until the contestant gets 11 identical answers to determine if the prize is behind this door. According to this method,

Assume the host lies $n$ times, which means the contestant has asked $11+n$ times so far, and the host can lie at most $10-n$ more times. Now, ask the host if the prize is behind the right door. Note that the host can only lie $10-n$ times, so the contestant needs to ask at most $2(10-n)+1=20-2n+1$ times to determine if the prize is behind this door.
Thus, the contestant has asked a total of $32-n$ times.
Therefore, 32 is the minimum number of questions required.
Next, we show that 31 questions cannot determine the prize's location.
At the beginning, the contestant asks about both doors up to 10 times, and the host answers "no." For convenience, assume the left door is the first door the contestant asks about on the 11th question (the other case is similar). Consider the scenario where the prize is not behind the left door.

At this point, the host can ensure that after 31 answers, the contestant cannot determine whether the prize is behind the middle or the right door.

When asked about the left door, the host answers "no" (the host always tells the truth about the left door). For the right door, the host answers "no" until the tenth time it is asked, and then answers "yes" for the next ten questions. Since the contestant can ask at most 20 questions about the right door (the first 11 questions were about the left door), and the host can lie at most 10 times, in both cases where the prize is behind the middle door or the right door, the host gives 31 identical answers. Therefore, the contestant cannot determine the prize's location.