一次数学竞赛共有 100 人参加, 其中 37 人做错了第一道题, 52 人做错了第二道题, 第一道题和第二道题都做对的 有 29 人。这两道题都做错的有多少人？---**Extracted Content:**
**Question 12**
**Question Stem:**
一次数学竞赛共有 100 人参加, 其中 37 人做错了第一道题, 52 人做错了第二道题, 第一道题和第二道题都做对的 有 29 人。这两道题都做错的有多少人？
**Translation of Question Stem (for clarity, not part of the required output but useful for understanding):**
A math competition had a total of 100 participants. Among them, 37 people made a mistake on the first problem, 52 people made a mistake on the second problem, and 29 people answered both the first and second problems correctly. How many people made mistakes on both problems?
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Let's solve this step by step. We have a math competition with 100 participants. 37 people made a mistake on the first problem, 52 people made a mistake on the second problem, and 29 people got both problems correct. We need to find how many people made mistakes on both problems.
To solve this problem systematically, we'll use set theory. Let U be the universal set of all 100 participants. Set A represents people who made mistakes on problem 1, containing 37 people. Set B represents people who made mistakes on problem 2, containing 52 people. The intersection A intersect B represents people who made mistakes on both problems, which is what we need to find. The complement A complement intersect B complement represents people who got both problems correct, which is 29 people.
Now let's visualize this problem using a Venn diagram. We have two overlapping circles representing people who made mistakes on each problem. This creates four distinct regions: people who got both problems correct, people who made mistakes on problem 1 only, people who made mistakes on problem 2 only, and people who made mistakes on both problems. We know 29 people got both correct, 37 total made mistakes on problem 1, and 52 total made mistakes on problem 2. Our goal is to find how many made mistakes on both problems.
Now let's use the complement rule to find a key insight. If 29 people got both problems correct, this means 29 people are in the complement of both mistake sets. Using the complement rule, the number of people who made at least one mistake equals 100 minus 29, which is 71 people. In set notation, the absolute value of A union B equals 71. This gives us a crucial piece of information for solving the problem.
Now we can apply the inclusion-exclusion principle to find our answer. The formula states that the size of A union B equals the size of A plus the size of B minus the size of A intersect B. Substituting our known values: 71 equals 37 plus 52 minus A intersect B. This simplifies to 71 equals 89 minus A intersect B. Solving for A intersect B, we get A intersect B equals 89 minus 71, which equals 18. Therefore, 18 people made mistakes on both problems.