AMC12 2022 A

AMC12 2022 A · Q9

AMC12 2022 A · Q9. It mainly tests Systems of equations, Logic puzzles.

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?

万圣节有$31$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；有些交替说谎和说真话。交替者任意选择第一个回答（谎言或真话），但后续每个陈述与其前一个的真值相反。校长按此顺序向每个人问了同样三个问题。 “你是说真话者吗？”校长给回答“是”的$22$个孩子每人一块糖果。 “你是交替者吗？”校长给回答“是”的$15$个孩子每人一块糖果。 “你是说谎者吗？”校长给回答“是”的$9$个孩子每人一块糖果。校长总共给了多少块糖果给总是说真话的孩子？

(A) 7 7

(B) 12 12

(D) 27 27

(E) 31 31

Answer

Correct choice: (A)

正确答案：（A）

Solution

Note that: - Truth-tellers would answer yes-no-no to the three questions in this order. - Liars would answer yes-yes-no to the three questions in this order. - Alternaters who responded truth-lie-truth would answer no-no-no to the three questions in this order. - Alternaters who responded lie-truth-lie would answer yes-yes-yes to the three questions in this order. Suppose that there are $T$ truth-tellers, $L$ liars, and $A$ alternaters who responded lie-truth-lie. The conditions of the first two questions imply that \begin{align*} T+L+A&=22, \\ L+A&=15. \end{align*} Subtracting the second equation from the first, we have $T=22-15=\boxed{\textbf{(A) } 7}.$ Remark The condition of the third question is extraneous. However, we know that $A=9$ and $L=6,$ so there are $9$ alternaters who responded lie-truth-lie, $6$ liars, and $9$ alternaters who responded truth-lie-truth from this condition.

注意： - 说真话者对三个问题的回答顺序为是-否-否。 - 说谎者对三个问题的回答顺序为是-是-否。 - 以真-谎-真回答的交替者对三个问题的回答顺序为否-否-否。 - 以谎-真-谎回答的交替者对三个问题的回答顺序为是-是-是。设 $T$ 为说真话者人数，$L$ 为说谎者人数，$A$ 为以谎-真-谎回答的交替者人数。前两个问题的条件给出 \begin{align*} T+L+A&=22, \\ L+A&=15. \end{align*} 第一式减去第二式，得 $T=22-15=\boxed{\textbf{(A) } 7}$。备注第三个问题的条件是多余的。然而，从该条件知 $A=9$，$L=6$，从而有$9$个以谎-真-谎回答的交替者、$6$个说谎者，以及$9$个以真-谎-真回答的交替者。

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