Problem 7 · AMC 8 Stretch
Core
Counting & Probability
Number Theory
account-for-all-possibilitiesorganizing-datapattern-recognition
Now use a 10-inch wire instead of 9, again bent at two inch marks to make a triangle. Will there be MORE choices, FEWER, or the SAME number as the 9-inch wire? Find the exact number of bending-point choices.
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Answer: 6 choices (fewer than the 9-inch wire's 10)
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Hint 1 of 4
Don't just guess 'longer means more!' Test it. List the whole-number side triples that add to 10 and obey the triangle rule.
Still stuck? Show hint 2 →
Hint 2 of 4
No side can be 5 or more (since 5 is half of 10, and a side that big can't be beaten by the other two). So all sides are 4 or less and add to 10.
Still stuck? Show hint 3 →
Hint 3 of 4
The only valid shapes are 2-4-4 and 3-3-4. Now count the actual bending-point pairs for each.
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Approach: List valid triangles, then count bending-point pairs
- For sides adding to 10 with the triangle rule, every side must be less than 5 (a side of 5 or more couldn't be beaten by the rest). The only whole-number triangles are 2-4-4 and 3-3-4, both isosceles — no scalene triangle fits.
- Count the bending-point pairs: 2-4-4 comes from {2,6}, {4,6}, {4,8}; and 3-3-4 comes from {3,6}, {3,7}, {4,7}.
- That's \(3 + 3 = 6\) choices, all isosceles — fewer than the 10 choices for the 9-inch wire.
- So the answer is 6: a longer wire does not always mean more triangles. Always check, don't assume!
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