Problem 11 · 2025 Math Kangaroo
Medium
Geometry & Measurement
caseworkspatial-reasoning
Robert wants to choose four points in such a way that the distances between any two of them are different. Which one of the points A, B, C, D or E must he remove?
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Answer: D — D
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Hint 1 of 3
Read each point's grid coordinates, then look for repeated distances rather than computing all ten exactly.
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Hint 2 of 3
Several pairs share the length \(\sqrt5\); find the one point common to the offending pairs.
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Hint 3 of 3
Remove that point and check the remaining six distances are all different.
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Approach: find the repeated length and the point it shares
- With \(A(0,3), B(1,3), C(2,2), D(2,1), E(0,0)\), the pairs \(AC, BD, DE\) all have length \(\sqrt5\), so a duplicate length is the obstacle.
- Point \(D\) appears in two of those equal pairs (\(BD\) and \(DE\)); the four remaining points \(A,B,C,E\) give distances \(1,\sqrt2,\sqrt5,2\sqrt2,3,\sqrt{10}\), all distinct.
- So Robert must remove point D, choice (D).
Mark:
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