Problem 21 · 2025 Math Kangaroo
Stretch
Geometry & Measurement
caseworksum-constraint
Four circular discs with radii \(r_1\), \(r_2\), \(r_3\) and \(r_4\) have their centres at the points \((0\,|\,0)\), \((1\,|\,0)\), \((3\,|\,0)\) and \((6\,|\,0)\). The discs may touch each other but may not overlap. What is the largest possible value of \(r_1 + r_2 + r_3 + r_4\)?
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Answer: B — 4
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Hint 1 of 2
Touching discs give one equation each: the sum of two radii equals the gap between their centres.
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Hint 2 of 2
Neighbouring constraints r₁+r₂ ≤ 1 and r₃+r₄ ≤ 3 already cap the total at 4.
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Approach: bound the sum using nearest-neighbour gaps
- Discs at 0 and 1 give r₁ + r₂ ≤ 1; discs at 3 and 6 give r₃ + r₄ ≤ 3.
- Adding: r₁+r₂+r₃+r₄ ≤ 4, and r₁ = 1, r₄ = 3 (others 0) achieves it.
- Largest possible sum = 4.
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