Score: $600$ points
E869120 is initially standing at the origin $(0, 0)$ in a two-dimensional plane.
He has $N$ engines, which can be used as follows:
He wants to go as far as possible from the origin. Let $(X, Y)$ be his final coordinates. Find the maximum possible value of $\sqrt{X^2 + Y^2}$, the distance from the origin.
Input is given from Standard Input in the following format:
$N$ $x_1$ $y_1$ $x_2$ $y_2$ $:$ $:$ $x_N$ $y_N$
Print the maximum possible final distance from the origin, as a real value. Your output is considered correct when the relative or absolute error from the true answer is at most $10^{-10}$.
3 0 10 5 -5 -5 -5
10.000000000000000000000000000000000000000000000000
The final distance from the origin can be $10$ if we use the engines in one of the following three ways:
The distance cannot be greater than $10$, so the maximum possible distance is $10$.
5 1 1 1 0 0 1 -1 0 0 -1
2.828427124746190097603377448419396157139343750753
The maximum possible final distance is $2 \sqrt{2} = 2.82842...$. One of the ways to achieve it is:
5 1 1 2 2 3 3 4 4 5 5
21.213203435596425732025330863145471178545078130654
If we use all the engines in the order $1 \rightarrow 2 \rightarrow 3 \rightarrow 4 \rightarrow 5$, we will end up at $(15, 15)$, with the distance $15 \sqrt{2} = 21.2132...$ from the origin.
3 0 0 0 1 1 0
1.414213562373095048801688724209698078569671875376
There can be useless engines with $(x_i, y_i) = (0, 0)$.
1 90447 91000
128303.000000000000000000000000000000000000000000000000
Note that there can be only one engine.
2 96000 -72000 -72000 54000
120000.000000000000000000000000000000000000000000000000
There can be only two engines, too.
10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
148.660687473185055226120082139313966514489855137208