Problem:
Score : 600 points

Problem Statement

We have an integer sequence A of length N, where A_1 = X, A_{i+1} = A_i + D (1 \leq i < N ) holds.

Takahashi will take some (possibly all or none) of the elements in this sequence, and Aoki will take all of the others.

Let S and T be the sum of the numbers taken by Takahashi and Aoki, respectively. How many possible values of S - T are there?

Constraints

-10^8 \leq X, D \leq 10^8

1 \leq N \leq 2 \times 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N X D

Output

Print the number of possible values of S - T.

Sample Input 1

3 4 2

Sample Output 1

8

A is (4, 6, 8).

There are eight ways for (Takahashi, Aoki) to take the elements: ((), (4, 6, 8)), ((4), (6, 8)), ((6), (4, 8)), ((8), (4, 6))), ((4, 6), (8))), ((4, 8), (6))), ((6, 8), (4))), and ((4, 6, 8), ()).

The values of S - T in these ways are -18, -10, -6, -2, 2, 6, 10, and 18, respectively, so there are eight possible values of S - T.

Sample Input 2

2 3 -3

Sample Output 2

2

A is (3, 0). There are two possible values of S - T: -3 and 3.

Sample Input 3

100 14 20

Sample Output 3

49805

Solution in Rust (1.15.1) (judge verdict: Accepted)

use std::cmp::max;
use std::collections::BTreeMap;

fn main() {
    let (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());
    let n: i64 = sc.read();
    let x: i64 = sc.read();
    let d: i64 = sc.read();

    if d == 0 {
        if x == 0 {
            println!("1");
        } else {
            println!("{}", n + 1);
        }
        return;
    }

    let (x, d) = if d > 0 { (x, d) } else { (-x, -d) };

    let mut map = BTreeMap::new();
    map.insert(0, vec![(0, 0)]);
    for i in 1..(n + 1) {
        let add = (i * x) / d;
        let from = (i - 1) * i / 2 + add;
        let to = (n - 1 + n - i) * i / 2 + add;
        let mut m = (i * (x % d)) % d;
        if m < 0 {
            m += d;
        }
        map.entry(m).or_insert(Vec::new()).push((from, to));
    }

    let mut ans = 0;
    for (_, mut v) in map.into_iter() {
        v.sort();
        let (mut cur_from, mut cur_to) = v[0];
        for (from, to) in v.into_iter() {
            if cur_to < from {
                ans += cur_to - cur_from + 1;
                cur_from = from;
                cur_to = to;
            } else {
                cur_to = max(cur_to, to);
            }
        }

        ans += cur_to - cur_from + 1;
    }

    println!("{}", ans);

    // S = zX + yD
    // 0+1+2+...+(z-1) <= y <= (N-z) + (N-z+1) + ... + (N-1)
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> IO<R, W> {
        IO(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: std::ops::Deref<Target = str>>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}