44 is wrong. It should be 42. Dyeing is also 42.

Method 1: 42

The number of people who are right in the first question plus those who are right in the second question, the number of people who are right in the third question plus those who are right in the fourth question exceeds 120, so there are definitely more people who are right in the fifth question, so it can be assumed that the people who are right in the fifth question are right in all questions.

In this way, the total number of people is 120-35=85, and the number of people who do the right questions is 6 1, 48, 39, 3 1 respectively.

Then, the number of people who have done the first question and the second question is 6 1+48-85=24, the number of people who have only done the first question is 37, the number of people who have only done the second question is 24, and the total number of people who have done the third and fourth questions is less than 85, so make sure that they do not intersect each other, because the intersection will inevitably produce more prizes. At the same time, the fewer people answer both questions correctly, the better, at least 3 1-24=7, 39-37 = 2 (including 2 out of 7).

So the number of winners is 35+7=42.

Dye

Solution: Number these 120 people as P 1, P2, …, P 120 respectively.

It is regarded as 120 point on the number axis, and Ak is used to indicate the group of 120 people who did not answer the K question correctly.

|Ak| is the number of people in this group, k=l, 2, 3, 4, 5,

Then |A 1|=24, |A2|=37, |A3|=46, |A4|=54, |A5|=85,

Give the above five groups five colors respectively,

If someone doesn't answer question k correctly,

This will mean that this person is recognized by K color, k = 1, 2, 3, 4, 5,

The question becomes, how many dots can you dye at least three colors at most?

Because | a1| +| a2 |+a3 |+| a4 |+| a5 | = 246,

Therefore, there are no more than three spots.

246

three

=82,

The upper right picture is the best dyeing method that meets the conditions.

That is, 85 points P 1, P2, …, P85 are dyed with the fifth color;

Points P 1, P2, …, P37 are dyed with the second color;

46 points P38, P39, …, P83 are dyed into the fourth color;

Dots P 1, P2, …, P24 are dyed with the first color;

54 points P25, P26, …, P78 are dyed in the third color;

So there are at most 78 dots dyed in three colors.

So there are at least 42 points with no more than two colors.

That is to say, there are at least 42 winners (each person answers at most two wrong questions and at least three right questions, such as P79, P80, …, P 120).