Difference between revisions of "2022 AMC 10B Problems/Problem 11"
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| + | ==Problem== | ||
| + | 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"? | ||
| + | <math>\textbf{(A) }</math> All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS. | ||
| + | |||
| + | <math>\textbf{(B) }</math> No school that sold more T-shirts than Euclid HS is bigger than Euclid HS. | ||
| + | |||
| + | <math>\textbf{(C) }</math> All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS. | ||
| + | |||
| + | <math>\textbf{(D) }</math> All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS. | ||
| + | |||
| + | <math>\textbf{(E) }</math> All schools smaller than Euclid HS sold more T-shirts than Euclid HS. | ||
| + | |||
| + | ==Solution== | ||
| + | Let <math>B</math> denote a school that is bigger than Euclid HS, and <math>M</math> denote a school that sold more T-shirts than Euclid HS. | ||
| + | |||
| + | It follows that <math>\neg B</math> denotes a school that is not bigger than Euclid HS, and <math>\neg M</math> 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 | ||
| + | |||
| + | ~MRENTHUSIASM | ||
| + | |||
| + | == See Also == | ||
| + | {{AMC10 box|year=2022|ab=B|num-b=10|num-a=12}} | ||
| + | {{MAA Notice}} | ||
Revision as of 19:45, 17 November 2022
Problem
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"?
All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.
No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.
All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.
All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.
All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
Solution
Let 
denote a school that is bigger than Euclid HS, and 
denote a school that sold more T-shirts than Euclid HS.
It follows that 
denotes a school that is not bigger than Euclid HS, and 
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
~MRENTHUSIASM
See Also
| 2022 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
