Cutting Bamboo

The interior of a segment meets $y=0$ exactly when its endpoint Y-coordinates have opposite signs. With the same sign the entire segment stays on one side; with one zero the intersection is only an endpoint. Both cannot be zero by the constraints. Count precisely the opposite-sign pairs. This takes $O(\sum N)$ time and $O(1)$ auxiliary space. The naive product test $Y_{1,i}Y_{2,i}<0$ also works. Its product has magnitude at most $10^{12}$, so use 64-bit integers.

Solution written by GPT6.1