AMC 8 1987 Problem 24
AMC8 1987 年第 24 题
特殊数与不定方程★★★★☆
A multiple choice examination consists of 20 questions. The scoring is +5 for each correct answer, -2 for each incorrect answer, and 0 for each unanswered question. John's score on the examination is 48 . What is the maximum number of questions he could have answered correctly?
一项选择题考试包含 20 道题。评分规则为：每答对一题 +5 分，每答错一题 -2 分，未答题 0 分。John 的考试分数为 48。他最多可能答对了多少道题？
- A.9
- B.10
- C.11
- D.12
- E.16
正确答案 · Correct Answer
D
解析 · Solution
解法1
常规解法 · 设元放缩
设答对 题，则答错至多 20-c 题，得分至少 5c-2(20-c)=7c-40。由 得 ，故 。而 c=12 可行：答对 12、答错 6，得 60-12=48。
巧解 · 试值 + 整除
要 5c-2w=48 且 。试 c=12：w=6，，成立；c=13：w=8.5 非整数，不行，更大只会更差。故最多 12。
Solution 1
Elementary · set a variable and bound
Let be the number correct; then the number wrong is at most 20-c, so the score is at least 5c-2(20-c)=7c-40. From , , so . And c=12 works: 12 right and 6 wrong give 60-12=48.
Slick · test values + divisibility
We need 5c-2w=48 with . Testing c=12: w=6 and , valid; c=13 gives w=8.5, impossible, and larger is worse. So the maximum is 12.
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