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2022 AMC 10B Problem 11

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2022 AMC 10B Problem 11AMC 10 Step by Step
Name:                             Date: September 13, 2026
Logic Puzzles · difficulty 2 of 5 · about 2 min · position P11-15

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"?

  1. A)

    AllschoolssmallerthanEuclidHSsoldfewerT−shirtsthanEuclidHS.All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.

  2. B)

    NoschoolthatsoldmoreT−shirtsthanEuclidHSisbiggerthanEuclidHS.No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.

  3. C)

    AllschoolsbiggerthanEuclidHSsoldfewerT−shirtsthanEuclidHS.All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.

  4. D)

    AllschoolsthatsoldfewerT−shirtsthanEuclidHSaresmallerthanEuclidHS.All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.

  5. E)

    AllschoolssmallerthanEuclidHSsoldmoreT−shirtsthanEuclidHS.All schools smaller than Euclid HS sold more T-shirts than Euclid HS.

Answer

B

Key insight

Rewrite the statement as 'if a school is bigger, then it did not sell more' and take the contrapositive: 'if a school sold more, then it is not bigger'.

Solution

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

An implication is equivalent to its contrapositive: if MM then not BB. 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 (B) No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.\boxed{\textbf{(B)}\ \text{No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.}}.

Why this works

"No XX is YY" is the implication "X⇒not YX \Rightarrow \text{not } Y," and swapping the roles of XX and YY (the contrapositive) gives an equivalent statement. The only other trap is the negation: "not more than" means "less than or equal to," never "fewer than." Convert wordy statements into ⇒\Rightarrow form before comparing.

Alternative approach

Build a tiny world: two schools, Euclid selling 100100 shirts and a bigger school selling exactly 100100. The original statement is true. Choice (C) is false in this world, so it is not equivalent; choice (B) is true. Worlds with a smaller school selling more shirts kill (A) and (E), and one with a same-size school selling fewer kills (D).

The trap

Choosing (C), which replaces 'did not sell more' with 'sold fewer' and so wrongly rules out ties.

Common mistakes

  • Choosing (C), which replaces 'did not sell more' with 'sold fewer' and so wrongly rules out ties.
  • Picking (A) or (D) by negating both parts of the implication (the inverse), which is not equivalent to the original.

Techniques

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