2019 AMC 12A Problems/Problem 25
Problem
Let be a triangle whose angle measures are exactly , , and . For each positive integer define to be the foot of the altitude from to line . Likewise, define to be the foot of the altitude from to line , and to be the foot of the altitude from to line . What is the least positive integer for which is obtuse?
Solution
For all nonnegative integers , let , , and .
Note that quadrilateral is cyclic since ; thus, . By a similar argument, . Thus, . By a symmetric argument, and .
Therefore, for any positive integer , we have (identical recurrence relations can be derived for and ). To find an explicit form for this recurrence, we guess that the constant term is related exponentially to (and the coefficient of is ). Hence, we let . We will solve for , , and by iterating the recurrence to obtain , , and . Letting respectively, we have
Subtracting from , we have , and subtracting from gives . Dividing these two equations gives , so . Substituting back, we get and . Hence, for all positive integers , , and the same holds for and .
The problem asks for the smallest such that either , , or is greater than . WLOG, let , , and . Thus, for all , , and . Solving for the smallest possible value of in each sequence, we find that gives . Therefore, the answer is .
-soyamyboya
See Also
2019 AMC 12A (Problems • Answer Key • Resources) | |
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