Problem 20 · 2018 Math Kangaroo
Medium
Algebra & Patterns
substitution
11 points are marked from left to right on a straight line, and their distances are recorded. The sum of the distances from the first point to every other point is 2018. The sum of all distances from the second point to every other point, including the first point, is 2000. What is the distance between the first and the second point?
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Answer: B — 2
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Hint 1 of 2
Compare the two distance-sums term by term, lining up the same far points.
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Hint 2 of 2
Every far point (from the 3rd on) contributes the gap between point 1 and point 2 to the difference.
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Approach: subtract the two sums to isolate the first gap
- From point 2's sum, the distance to point 1 is the gap d; the distances to points 3..11 are each shorter by d than from point 1.
- So the sums differ by 9d (nine points beyond the second): 2018 − 2000 = 9d.
- Thus 9d = 18 and the first gap is d = 2.
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