AMC10 2022 B

AMC10 2022 B · Q11

AMC10 2022 B · Q11. It mainly tests Logic puzzles.

All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?

一个大型学区内所有高中都参与了销售T恤的筹款活动。以下哪个选项与陈述“没有比Euclid高中更大的学校卖出的T恤比Euclid高中多”在逻辑上是等价的？

(A) All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS. All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.

(B) No school that sold more T-shirts than Euclid HS is bigger than Euclid HS. No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.

(C) All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS. All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.

(D) All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS. All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.

(E) All schools smaller than Euclid HS sold more T-shirts than Euclid HS. All schools smaller than Euclid HS sold more T-shirts than Euclid HS.

Answer

Correct choice: (B)

正确答案：（B）

Solution

Let $B$ denote a school that is bigger than Euclid HS, and $M$ denote a school that sold more T-shirts than Euclid HS. It follows that $ eg B$ denotes a school that is not bigger than Euclid HS, and $ eg M$ denotes a school that did not sell more T-shirts than Euclid HS. Converting everything to conditional statements (if-then form), the given statement becomes \[B\implies eg M.\] Its contrapositive is $M\implies eg B,$ which is $\boxed{\textbf{(B)}}.$ Note that "not bigger than" does not mean "smaller than", and "not selling more" does not mean "selling fewer". There is an equality case. Therefore, none of the other answer choices is equivalent to $B\implies eg M.$

设$B$表示比Euclid高中更大的学校，$M$表示卖出比Euclid高中更多的T恤的学校。由此，$ eg B$表示不比Euclid高中大的学校，$ eg M$表示没有卖出比Euclid高中更多的T恤的学校。将一切转换为条件语句（if-then形式），给定陈述变为\[B\implies eg M.\] 其逆否命题是$M\implies eg B$，即\boxed{\textbf{(B)}}。注意“不比...大”不意味着“比...小”，“没有卖出更多”不意味着“卖出更少”。存在相等情况。因此，其他选项都不等价于$B\implies eg M$。

Topics

LM.001 Logic puzzles