AMC 8 2020 Problem 22
AMC8 2020 年第 22 题
奇偶性★★★☆☆
When a positive integer is fed into a machine, the output is a number calculated according to the rule shown below. For example, starting with an input of N = 7, the machine will output . Then if the output is repeatedly inserted into the machine five more times, the final output is 26: . When the same 6-step process is applied to a different starting value of , the final output is 1. What is the sum of all such integers ?
当把一个正整数 N 输入一个机器，输出的是一个根据下面规则计算得到的数字：
例如，一开始输入的是 N=7，那么机器将输出接着如果把输出再输入机器这样继续重复 5 次，那么最后的输出将是 7→22→11→34→ 17 → 52 → 26。当把上述同样的 6 步过程再作用于另一个不同的初始值 N，最终的输出是 1，那么 N 的所有可能值之和是多少?
- A.73
- B.74
- C.75
- D.82
- E.83
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
Working backwards from the number 1 1, as shown below, there are 4 4 possible starting values of 𝑁: 1, 8, 10 N:1,8,10, and 64 64. The sum of these values is 1 + 8 + 10 + 64 = 83 1+8+10+64=83. 𝑁 𝑜 𝑡 𝑒 Note: The Collatz Conjecture states that the number 1 1 will appear eventually no matter which positive integer 𝑁 N is chosen as the starting value. For example if the 𝑁 = 7 N=7 sequence is extended past 26 26, the outputs will be 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1. 26→13→40→20→10→5→16→8→4→2→1. It is not known whether this conjecture is true. Answer: 𝐸 E .
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