Problem 14 · 2023 Math Kangaroo
Stretch
Geometry & Measurement
casework
In a three-sided pyramid all side lengths are integers. Four of the side lengths can be seen in the diagram. What is the sum of the two remaining side lengths?
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Answer: C — 11
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Hint 1 of 3
Each of the four triangular faces must obey the triangle inequality with whole-number sides.
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Hint 2 of 3
The two unknown edges each sit in two faces; intersect the allowed ranges from those faces.
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Hint 3 of 3
The two visible edges of a face squeeze the third edge into a narrow whole-number window.
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Approach: intersect the triangle-inequality ranges for each missing edge
- The shown edges are 7 and 2 on one face and 3 and 4 on another; the two missing edges close up the remaining faces.
- The missing edge in the 7-2 region must satisfy \(5 < e < 9\), and the same edge in the 3-4 region must satisfy \(1 < e < 7\), forcing it to be 6.
- The other missing edge must satisfy \(4 < e < 10\) and \(2 < e < 6\), forcing it to be 5, so the two missing edges sum to \(6 + 5 = \mathbf{11}\).
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