Editorial for CTU Open Contest 2017 - Go Northwest! - Olympiads Online Judge

Editorial for CTU Open Contest 2017 - Go Northwest!

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We need to count the number of ways to select two (not necessarily distinct!) points such that when we connect them by a line, the angle between the line and horizontal axis is $45^\circ$ ~45^\circ~
How?

$\text{[math]}$

The coordinates of points can help us to index diagonals to find out how many points lie on the same diagonal
The northwest diagonals can be indexed by $x+y$ ~x+y~
The northeast diagonals can be indexed by $x-y$ ~x-y~
For each diagonal that contains $x$ ~x~ points, the number of ways to select two points to form a line is $x \times (x-1)$ ~x \times (x-1)~
That leaves us to sum the result over all diagonals and divide it by number of all options $N^2$ ~N^2~
Total running time: $\mathcal O(N)$ ~\mathcal O(N)~ or $\mathcal O(N \log N)$ ~\mathcal O(N \log N)~ depending on the indexing of diagonals

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