Given a bitonic sequence $a_1, a_2, \ldots, a_n$ , calculate the median in $O(\log n)$
First you use an algorithm to find the bitonic point, that is the point where the sequence swaps from being non-increasing to non-decreasing. This is possible with this algorithm in $O(\log(n))$. Now you split the sequence apart at this point (virtually, you don't really make any copies since that would cost $O(n)$). You can now use this algorithm to find the median of two sorted sequences in $O(\log(\min(u,v)))$. $u+v = n$ in our case since we split the original sequence.
Beware that you need to adjust the given algorithms since the first one uses the opposite definition of bitonic (first increasing the decreasing) and the second one assumes that both arrays are sorted in the same direction (in our case one is in reverse).