Problem 8 · 2015 AMC 8
Medium
Geometry & Measurement
perimeter
What is the smallest whole number larger than the perimeter of any triangle with a side of length 5 and a side of length 19?
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Answer: D — 48.
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Hint 1 of 2
What makes the perimeter big is the third side, and it can't grow without limit: the triangle inequality says it must stay shorter than the other two sides combined. Cap the third side and you cap the perimeter.
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Hint 2 of 2
The cap is a strict 'less than' — the perimeter gets as close to it as you like but never touches it. So the smallest whole number larger than every possible perimeter is exactly that boundary value (the supremum it can't reach).
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Approach: the triangle inequality caps the perimeter; find that ceiling
- The third side s must satisfy s < 5 + 19 = 24 (it also needs s > 14, but the upper bound is what limits the perimeter).
- Perimeter P = 5 + 19 + s < 5 + 19 + 24 = 48, and this is strict — P can creep up toward 48 (e.g. s = 23.9 gives P = 47.9) but never reach it.
- Since every perimeter is below 48 yet can exceed any number under 48, the smallest whole number larger than all of them is 48.
- Why this transfers: 'smallest integer greater than a quantity that approaches but never hits a bound' lands you exactly on the bound itself — the strict inequality is doing the real work.
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