The problem asks for minimum number of lines passing through origin which covers all points.
We can make $N$ lines in which each $i^{th}$ line covers only $i^{th}$ point. Since the line must pass through origin so we have two points, except when $(x_i, y_i) = (0, 0)$.
Now the only thing left is to unique how many distinct lines exist among these.
For example: $x = 2 \cdot y$ and $2 \cdot x = 4 \cdot y$ represent same line.
We can check number of distinct lines by checking number of distinct slopes.
There will be precision issues if you use double to store points or slope. Same precision issue will be in python. Try doing print(1 == 1 + 1e-16). One other mistake participants did was division by $0$.
We tried a lot but couldn't make a case for which long double fails due to precision issues.
Instead of storing $(\frac{y}{x})$, we store $(x, y)$ representing denominator and numerator of fraction in it's simplest form respectively.
Simplest Form of $(2, 4)$ is $(1, 2)$. To make fraction in it's simplest form simply make $x = \frac{x}{gcd(x, y)}$ and $y = \frac{y}{gcd(x, y)}$.
The only case that is left to handle is $(1, -2)$ and $(-1, 2)$. To handle that we can fix one out of numerator and denominator which will be negative (if one of the two is negative). See Code below for implementation details.