Problem 12 · 2009 Math Kangaroo
Medium
Logic & Word Problems
off-by-onesum-constraint
In a group of 2009 kangaroos each one is either light or dark. The smallest of the light kangaroos is bigger than exactly 8 dark kangaroos. One light one is bigger than exactly 9 dark ones, another light one is bigger than exactly 10 dark ones, and so on. Exactly one light kangaroo is bigger than all dark kangaroos. How many light kangaroos are there?
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Answer: B — 1001
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Hint 1 of 2
Line up the light kangaroos by size and read off how many darks each one beats.
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Hint 2 of 2
The counts 8, 9, 10, … rise by one per light kangaroo until one beats every dark.
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Approach: match each light kangaroo to its ‘beats-this-many-darks’ count
- The k-th smallest light kangaroo beats exactly 7 + k dark kangaroos.
- The largest light beats all D darks, so 7 + L = D, where L lights and D darks total 2009.
- Then L + (L + 7) = 2009 gives L = 1001.
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