AMC 8 2011 Problem 21
AMC8 2011 年第 21 题
整除与因数★★★☆☆
Students guess that Norb's age is 24, 28, 30, 32, 36, 38, 41, 44, 47 , and 49 . Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?
学生猜测 Norb 的年龄是 24，28，30，32，36，38，41，44，47 和 49 岁。Norb 说：“你们中至少有一半猜的太低，有 2 个同学猜的跟实际年龄相差 1 岁，并且我的年龄是个质数”。
那么 Norb 多大？
- A.29
- B.31
- C.37
- D.43
- E.48
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
At least half the guesses are too low, so Norb's age must be greater than 36. If two of the guesses are off by one, then his age is in between two guesses whose difference is 2. It could be 31,37, or 48, but because his age is greater than 36 it can only be 37 or 48. Lastly, Norb's age is a prime number so the answer must be 37. Since two guesses are off by one, we know that both x+1 and x-1 are in the list where is the age of Norb. Now, we know that x+1 and x-1 are 28 and 30, 30 and 32, 36 and 38 and 47 and 49. From these values, we know that must be 29, 31, and 37. Since half of the guesses are too low, 24, 28, 30, 32, and 36 are all too low so we can eliminate all numbers in our list lesser than or equal to 36. Therefore, our list has only 37 left so the answer is 37. If his age is 29, then there won't be half of them who guessed less than his age. If his age is 31, then there still won't be half of them who guessed less than his age. If his age is 37, then there are half of them who guessed too low, half of them who guessed too high, and there are two people who guessed off by one. So, our answer is probably , but let's still check the other two choices. If his age is 43, then there won't be two people who were off by one. Finally, if his age is 48, then his age won't be a prime number. Therefore, our answer is 37.
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