Problem 19 · 2010 Math Kangaroo
Hard
Geometry & Measurement
careful-counting
What is the smallest number of straight lines with which a plane can be divided into exactly 5 regions?
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Answer: B — 4
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Hint 1 of 2
A new line adds 1 region for itself plus 1 for each line it crosses.
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Hint 2 of 2
To gain just one region at a time, keep the new line parallel — count how many parallels give 5 pieces.
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Approach: grow regions one at a time
- Parallel lines split the plane into (number of lines)+1 strips.
- Two lines reach at most 4 regions, and three lines cannot land on exactly 5.
- Four parallel lines give 4+1 = 5 regions, so the smallest count is 4.
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