Problem 29 · 2015 Math Kangaroo
Stretch
Logic & Word Problems
casework
Ten different numbers are written down. Each number which is equal to the product of the other nine numbers can then be underlined. What is the maximum amount of numbers that can be underlined?
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Answer: B — 2
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Hint 1 of 2
If a number equals the product of the other nine, that product is special; can many numbers be like that at once?
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Hint 2 of 2
Examine how two such numbers could coexist (think about signs and magnitudes) — you can get at most 2.
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Approach: bound how many can equal the others' product
- Suppose a number equals the product of the other nine; multiplying all ten then gives that number squared.
- Analysing the constraints, at most two of the ten numbers can satisfy this at once.
- Maximum underlined = 2.
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