2011 AMC 12B Problems/Problem 22
Problem
Let be a triangle with side lengths , , and . For , if and , and are the points of tangency of the incircle of to the sides , , and , respectively, then is a triangle with side lengths , and , if it exists. What is the perimeter of the last triangle in the sequence ?
Solution
Answer: (D)
Let , , and
Then , and
Then , ,
Hence:
Note that and for , I claim that it is true for all , assume for induction that it is true for some , then
Furthermore, the average for the sides is decreased by a factor of 2 each time.
So is a triangle with side length , ,
and the perimeter of such is
Now we need to find when fails the triangle inequality. So we need to find the last such that
For , perimeter is
See also
Identical problem to the 2011 AMC 10B Problems/Problem 25.
2011 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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