Difference between revisions of "2002 AMC 12P Problems/Problem 24"
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== Problem == | == Problem == | ||
| − | + | Let <math>ABCD</math> be a regular tetrahedron and Let <math>E</math> be a point inside the face <math>ABC.</math> Denote by <math>s</math> the sum of the distances from <math>E</math> to the faces <math>DAB, DBC, DCA,</math> and by <math>S</math> the sum of the distances from <math>E</math> to the edges <math>AB, BC, CA.</math> Then <math>\frac{s}{S}</math> equals | |
| − | <math> \ | + | <math> |
| + | \text{(A) }\sqrt{2} | ||
| + | \qquad | ||
| + | \text{(B) }\frac{2 \sqrt{2}}{3} | ||
| + | \qquad | ||
| + | \text{(C) }\frac{\sqrt{6}}{2} | ||
| + | \qquad | ||
| + | \text{(D) }2 | ||
| + | \qquad | ||
| + | \text{(E) }3 | ||
| + | </math> | ||
== Solution == | == Solution == | ||
Revision as of 00:05, 30 December 2023
Problem
Let 
be a regular tetrahedron and Let 
be a point inside the face 
Denote by 
the sum of the distances from to the faces 
and by 
the sum of the distances from to the edges 
Then 
equals
Solution
If 
, then 
. Since 
, 
must be 
to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of 
.
See also
| 2002 AMC 12P (Problems • Answer Key • Resources) | |
| Preceded by Problem 23 |
Followed by Problem 25 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
