2024 AIME I Problems/Problem 14
Problem
Let be a tetrahedron such that , , and . There exists a point inside the tetrahedron such that the distances from to each of the faces of the tetrahedron are all equal. This distance can be written in the form , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .
Solution 1
Notice that , , and , let , , , and . Then the plane has a normal
Hence, the distance from to plane , or the height of the tetrahedron, is
Each side of the tetrahedron has the same area due to congruency by "S-S-S", and we call it . Then by the volume formula for cones,
Solution by Quantum-Phantom
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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