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
See Also
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