Problem 10 · AMC 8 Stretch
Core
Logic & Word Problems
Geometry & Measurement
proof-by-contradictionparity
Is it possible to draw a single straight line that crosses every one of the \(999\) sides of a \(999\)-sided polygon? Explain why or why not.
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Answer: No — it is impossible (a parity argument, since 999 is odd)
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Hint 1 of 4
Suppose it IS possible, and look for something that goes wrong (this is proof by contradiction). A straight line splits the plane into two sides — call them 'left' and 'right'.
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Hint 2 of 4
Walk around the polygon vertex by vertex. Every time you cross the line, you switch from one side to the other. So crossing a side means the two endpoints of that side are on opposite sides of the line.
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Hint 3 of 4
If the line crosses ALL 999 sides, then every pair of neighboring vertices is on opposite sides — the vertices must perfectly alternate left, right, left, right... all the way around.
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Approach: Proof by contradiction using parity
- Suppose, for contradiction, that one straight line crosses all 999 sides. The line cuts the plane into two halves.
- Walk around the polygon from vertex to vertex. Crossing a side means stepping over the line, switching half-planes, so a crossed side has its two endpoints on opposite sides.
- If every one of the 999 sides is crossed, the vertices must alternate left, right, left, right around the whole loop and return to the start.
- Returning to the start after alternating only works with an even number of vertices. Since \(999\) is odd, perfect alternation around the loop is impossible — a contradiction. So no such line exists.
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