Problem 25 · 2020 Math Kangaroo
Stretch
Number Theory
factor-triplescasework
Julia wrote four positive integers, one at each vertex of a triangular-based pyramid. She calculated the sum of the numbers written on the vertices of one face and the products of the numbers written on the vertices of the other two faces, obtaining 15, 20 and 30, respectively. What is the highest possible value of the product of the four numbers?
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Answer: E — 120
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Hint 1 of 2
Each face uses three of the four vertices; two faces share an edge (two vertices).
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Hint 2 of 2
Pick vertex values matching the two products that share a pair, then check the sum-15 face.
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Approach: match shared-edge factor triples and maximise the product
- The faces with products 20 and 30 share two vertices; try the shared pair {1,5}.
- Then the third vertices are 4 (for 20) and 6 (for 30), giving values 1, 5, 4, 6.
- The remaining face 5+4+6 = 15 matches the required sum.
- The product of all four is 1·5·4·6 = 120.
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