Problem 28 · 2012 Math Kangaroo
Hard
Number Theory
careful-countingcasework
There are 30 chapters in a book. Each chapter has a different length, i.e. 1, 2, 3, …, 30 pages. Each chapter starts on a new page. The first chapter starts on page 1. At most how many chapters start on a page with an odd page number?
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Answer: E — 23
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Hint 1 of 2
A chapter keeps the next start's parity the same when its length is even, and flips it when odd.
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Hint 2 of 2
There are 15 even and 15 odd lengths — spend the even ones while you are on odd pages.
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Approach: track start-page parity
- Chapter 1 starts on page 1 (odd). An even-length chapter keeps the next start odd; an odd-length chapter flips the parity.
- Use all 15 even lengths first: starts of chapters 1 through 16 are all odd (16 odd starts). The remaining 14 odd-length chapters then alternate parity, adding 7 more odd starts.
- Maximum odd-page starts = 16 + 7 = 23.
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