Problem 9 · AMC 8 Stretch
Core
Geometry & Measurement
Logic & Word Problems
considering-extreme-casesaccount-for-all-possibilities
On the round Earth, from how many starting points can you walk exactly \(1\) mile south, then \(1\) mile east, then \(1\) mile north and end up right where you started? Describe ALL such points.
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Answer: Infinitely many: the North Pole plus infinitely many circles near the South Pole
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Hint 1 of 4
Look at the extreme spots on Earth: the poles. Try starting at the North Pole and check what happens.
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Hint 2 of 4
There is more than one answer! Think about the South Pole area. The 'east' walk goes around a circle of latitude. When would walking \(1\) mile east bring you back to where you started that leg?
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Hint 3 of 4
If your \(1\) mile south lands you on a tiny circle whose distance all the way around is exactly \(1\) mile, then walking \(1\) mile east loops you all the way around, back to the same spot. Then \(1\) mile north returns you to start.
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Approach: Considering extreme cases (the poles) and accounting for all possibilities
- North Pole: start there, walk \(1\) mile south, then \(1\) mile east along a latitude circle, then \(1\) mile north — back at the pole. It works.
- Near the South Pole: find the circle that is exactly \(1\) mile all the way around. Stand anywhere \(1\) mile NORTH of it. Walking south lands you on the little circle, walking \(1\) mile east loops you all the way around back to the same point, and walking north returns you to start. Every point on that bigger circle works — infinitely many.
- More possibilities: the little circle could instead be \(\frac{1}{2}\) mile around (the east walk loops it twice), or \(\frac{1}{3}\) mile around (three times), and so on for any whole number of loops.
- Full answer: the North Pole, plus — for every whole number \(n = 1, 2, 3, \dots\) — every point \(1\) mile north of the southern circle whose distance around is \(\frac{1}{n}\) mile. That is infinitely many starting points.
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