Problem 27 · 2015 Math Kangaroo
Stretch
Number Theory
sum-constraintcasework
On a straight line there are five points. Alex measures all the distances between every possible pair of points. He obtains in ascending order 2, 5, 6, 8, 9, k, 15, 17, 20 and 22. What is the value of k?
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Answer: E — 14
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Hint 1 of 2
Five points give 10 pairwise distances; the largest is the whole span and the smallest are the outer gaps.
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Hint 2 of 2
Fix the endpoints at 0 and 22, fit the inner points to match the distances, then read off the missing one.
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Approach: reconstruct the point positions from the distances
- Put the endpoints at 0 and 22, so the span 22 is the largest distance.
- The two smallest gaps, 2 and 5, sit at the ends, which places points at 2 and 17; matching the rest forces the last point to 8, giving coordinates 0, 2, 8, 17, 22.
- Their ten pairwise distances are 2, 5, 6, 8, 9, 14, 15, 17, 20, 22; the one missing from the given list is 14, so k = 14.
- So k = 14 (E).
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