Problem 9 · AMC 8 Stretch
Stretch
Geometry & Measurement
Counting & Probability
visual-representationlogical-reasoning
Mark 9 equally spaced dots, 0 through 8, around a circle. Dot 0 is where the wire's ends meet; pick 2 of the other 8 dots for the bends. The three arcs between your chosen dots are the three side lengths (they add to 9). There are \(\binom{8}{2} = 28\) ways to pick 2 dots. Of those 28, how many give a real triangle — one where NO arc is half the circle or more?
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Answer: 10 of the 28 dot-pairs (matching the 10 wire triangles)
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Hint 1 of 4
Each side of the wire triangle is one arc on the circle. The whole circle is 9 units. What does 'half the circle' equal in units?
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Hint 2 of 4
The triangle rule says each side must be SHORTER than the other two added together. Since all three add to 9, that means each side must be less than \(\tfrac92 = 4.5\).
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Hint 3 of 4
So 'each side < 4.5' is the same as 'each arc is less than half the circle.' Count how many of the 28 dot-pairs make all three arcs smaller than 4.5.
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Approach: Translate the triangle rule into 'every arc less than half the circle'
- A wire triangle uses dot 0 plus 2 chosen dots, so there are \(\binom{8}{2} = \tfrac{8 \times 7}{2} = 28\) ways to choose.
- For a real triangle each side must be shorter than the sum of the other two. Since the three sides always add to 9, that condition is simply: each side is less than \(\tfrac92 = 4.5\).
- Each side IS one arc of the 9-unit circle, so 'each side < 4.5' means 'each arc is less than half the circle.' When all three arcs are under half, the triangle drawn on the circle wraps around the center; if one arc reaches half or more, that side is too long and the wire can't close.
- Exactly 10 of the 28 dot-pairs keep every arc under 4.5 — the same 10 wire triangles found before. The other \(28 - 10 = 18\) have an arc of 5 or more and fail. So the answer is 10.
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