2014 AMC 8 Problem 23
Attempt Problem 23 of the 2014 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2014 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
Three members of the Euclid Middle School girls’ softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all -digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That’s funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of your two uniform numbers is today’s date.
What number does Caitlin wear?
Answer: A
Small hint:
Pair sums of two two-digit primes must be dates in a month.
Big hint:
Use the earlier-today-later order to determine which uniform number is smallest.
Solution:
Any pair-sum must be a date in the month, so it is at most . The possible sums of two two-digit primes that are at most are , , , and .
The three dates are earlier, today, and later, so they must be distinct and increasing. The only possible set of three distinct dates is , coming from the numbers , , and .
Bethany’s date is Ashley plus Caitlin, Ashley’s date is Bethany plus Caitlin, and Caitlin’s later date is Ashley plus Bethany. This means Caitlin’s number is the smallest of the three, namely .
Thus, A is the correct answer.
Problem 23 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
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- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
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