P4758 [CERC2014] Mountainous landscape

You travel through a scenic landscape consisting mostly of mountains – there are $n$ landmarks (peaks and valleys) on your path. You pause for breath and wonder: which mountain are you currently seeing on the horizon?  Formally: you are given a polygonal chain $P_1,P_2,\cdots,P_n$ in the plane. The $x$ coordinates of the points are in strictly increasing order. For each segment $P_i P_{i+1}$ of this chain, find the smallest index $j > i$, for which any point of $P_j P_{j+1}$ is visible from $P_i P_{i+1}$ (lies **strictly above** the ray $P_i \ P^{\rightarrow}_{i+1}$).

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