2015 UMO Problems/Problem 6


A triangular pyramid with apex $O$ and base $\triangle{ABC}$ has the property that the perimeter of $\triangle {ABC}$ is $84$. Additionally, one can place a cylinder of radius $4$ and height $10$ completely inside the pyramid such that one of its bases is in the same plane as $\triangle{ABC}$. What is the minimum possible height from $\triangle{ABC}$ to apex $O$? Show that this height is achievable.


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