Problem 24 · 2009 Math Kangaroo
Stretch
Geometry & Measurement
sum-constraintcasework
For how many whole numbers \(n\ge 3\) does there exist a convex polygon whose angles are in the ratio \(1:2:\cdots:n\)?
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Answer: B — 2
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Hint 1 of 2
Write the angles as 1x, 2x, …, nx and use the polygon’s angle-sum.
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Hint 2 of 2
Convexity needs every angle below 180°—especially the largest, nx—which limits n.
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Approach: angle-sum plus the convexity ceiling on the largest angle
- The angles sum to 180(n − 2), so x · n(n+1)/2 = 180(n − 2) and the largest angle is nx = 360(n − 2)/(n + 1).
- Convexity needs nx < 180, which simplifies to n < 5, so only n = 3 and n = 4 work.
- That is 2 values.
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