Question

万圣节时， $31$ 个孩子走进校长办公室要糖果。他们可分为三类：有些总是说谎，有些总是说真话，还有些真话和谎话交替说。交替者可以任意选择第一次回答是真话还是谎话，但之后每次回答的真假都与前一次相反。校长按顺序问每个人同样的三个问题。

“你是说真话者吗？”校长给每个回答“是”的孩子一颗糖，共有 $22$ 人回答“是”。

“你是交替者吗？”校长给每个回答“是”的孩子一颗糖，共有 $15$ 人回答“是”。

“你是说谎者吗？”校长给每个回答“是”的孩子一颗糖，共有 $9$ 人回答“是”。

校长总共给那些总是说真话的孩子多少颗糖？

On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.

“Are you a truth-teller?” The principal gave a piece of candy to each of the $22$ children who answered yes.

“Are you an alternater?” The principal gave a piece of candy to each of the $15$ children who answered yes.

“Are you a liar?” The principal gave a piece of candy to each of the $9$ children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?

$7$

$12$

$21$

$27$

$31$

答案：A

知识点：说真话者与说谎者逻辑推理

难度评级：1950

解答：

对于第一个问题，总是说真话者会回答“是”，总是说谎者也会回答“是”，第一次选择说谎的交替者同样会回答“是”；第一次选择说真话的交替者则回答“否”。记相应人数后可写成 $22 = t + l + a_l.$

对于第二个问题，总是说谎者会回答“是”；第一次选择说谎的交替者这轮必须说真话，也会回答“是”。其余两类回答“否”，所以 $15 = l + a_l.$

由此得到 $t = 7$ 。校长第一轮只给一直说真话的孩子糖果，因此总共只发给他们 $7$ 块糖。

所以正确答案是 A。

For the first question, the truth-tellers will respond yes, the liars will respond yes, and the alternaters who decided to lie first will say yes. The alternaters who decide to tell the truth first will say no. Denote this as $22 = t + l + a_l.$

For the second questions, the liars will respond yes, and the alternaters who decided to lie first will say yes (they are forced to tell the truth for this question). The truth-tellers will respond no, and the alternaters who told the truth first would lie this round, responding no. Denote this as $15 = l + a_l.$

From this, we get that $t = 7.$ The principal only gives candy to children who always tell the truth in the first round, therefore only giving them $7$ candies total.

Thus, A is the correct answer.

2022 AMC 10A 第 12 题

答案：A

难度评级：1950

解答：

其他年份的第 12 题