1997 AMC 12 Problem 22
Attempt Problem 22 of the 1997 AMC 12 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 1997 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
22.
Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had The absolute difference between the amounts Ashley and Betty had to spend was The absolute difference between the amounts Betty and Carlos had was between Carlos and Dick was between Dick and Elgin was and between Elgin and Ashley was How much did Elgin have?
Answer: E
Small hint:
Assign a sign to each successive difference around the five-person cycle
Big hint:
The signed differences must total zero before the five amounts can be summed
Solution:
The signed differences around the cycle have magnitudes and sum Thus one side of the sign split must total half of namely The only split is In one orientation, the amounts are and Their total is so and Reversing every sign would give not an integral Thus Elgin had and E is correct.
Problem 22 in other years
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