2022 AMC 10B Problem 11

P11-15~2 minPrint

Logic Puzzles Set up the equation/formula and compute; no special trick needed Test small/specific values or special cases to find or verify the answer

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"?

Show solution without answering

Key insight

The statement says a school cannot be both bigger than Euclid and a better seller than Euclid. Saying it in the other order gives the same meaning.

Solution

Read the statement as a rule about which schools cannot exist.

What the statement forbids. "No school bigger than Euclid HS sold more T-shirts than Euclid HS" means: there is no school that is both bigger than Euclid and sold more shirts than Euclid.
Say it the other way round. "Bigger and sold more" is the same pair of facts as "sold more and bigger". So an equally good way to say it is: no school that sold more shirts than Euclid is bigger than Euclid. That is exactly choice (B).
Why (C) is wrong. (C) says every bigger school sold fewer shirts. The original allows a bigger school that sold exactly as many shirts as Euclid. Example: Euclid sells $100$ shirts and a bigger school also sells $100$ . The original is true (nobody bigger sold more), but (C) is false. So they do not mean the same thing.
Why (A), (D), (E) are wrong. The original says nothing at all about smaller schools, so (A) and (E), which make claims about smaller schools, add information that is not there. (D) makes a claim about schools that sold fewer shirts, which the original also says nothing about. For example, a school of the same size as Euclid that sold fewer shirts makes (D) false but leaves the original true.

The answer is $\boxed{\textbf{(B)}\ \text{No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.}}$ .

Why this works

"No A is B" means nothing is both A and B, so it is the same as "No B is A". In logic this swap is called the contrapositive. The second lesson is about the word "not": "did not sell more" means "sold fewer or the same", so a choice that drops the "same" case says more than the original. To test a choice, invent a small example where one statement is true and the other is false.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Let $B$ mean "the school is bigger than Euclid" and $M$ mean "the school sold more T-shirts than Euclid." The given statement says: for every school, if $B$ then not $M$ .

An implication is equivalent to its contrapositive: if $M$ then not $B$ . In words: every school that sold more T-shirts than Euclid is not bigger than Euclid, i.e. "No school that sold more T-shirts than Euclid HS is bigger than Euclid HS." That is choice (B).

Check the others. (A) and (E) talk about smaller schools, about which the statement says nothing. (D) is the inverse-style statement about schools selling fewer shirts, again unrelated. (C) is close but says bigger schools sold fewer shirts; the original allows a bigger school to sell exactly as many, so (C) is strictly stronger and not equivalent.

The answer is $\boxed{\textbf{(B)}\ \text{No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.}}$ .

Review in the AoPS books

Intermediate Counting & Probability, Chapter 2: Sets and Logic (§2.4)