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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Globalization;
using System.IO;
using System.Text;
using System.Linq;

using E = System.Linq.Enumerable;

internal partial class Solver {
    public void Run() {
        var N = ni();
        var X = ni();
        var D = ni();
        cout.WriteLine(Solve(N, X, D));
    }

    internal static long SolveNaive(int n, long x, long d) {
        var a = new long[n];
        a[0] = x;
        for (int i = 1; i < n; i++) {
            a[i] = a[i - 1] + d;
        }
        var hashset = new HashSet<long>();
        for (int i = 0; i < (1 << n); i++) {
            var s = 0L;
            for (int j = 0; j < n; j++) {
                if (((1 << j) & i) != 0) {
                    s += a[j];
                }
            }
            hashset.Add(s);
        }
        return hashset.Count;
    }

    internal static long Solve(int n, long x, long d) {
        if (d == 0) {
            return x == 0 ? 1 : n + 1;
        }
        long lower = 0;
        long upper = 0;
        var segments = new Dictionary<long, List<Tuple<long, long>>>();

        segments[0] = new List<Tuple<long, long>> { Tuple.Create(0L, 0L) };

        for (int i = 1; i <= n; i++) {
            lower += i - 1;
            upper += n - i;

            // k*X + t*D
            var mod = (1L * i * x) % Math.Abs(d);
            if (mod < 0) mod += Math.Abs(d);
            var offset = (i * x - mod) / Math.Abs(d);
            List<Tuple<long, long>> list;
            if (!segments.TryGetValue(mod, out list)) {
                list = new List<Tuple<long, long>>();
                segments.Add(mod, list);
            }
            //new { mod, lower, upper, offset }.Dump();
            if (d > 0) {
                list.Add(Tuple.Create(lower + offset, upper + offset));
            } else {
                list.Add(Tuple.Create(-upper + offset, -lower + offset));
            }
        }

        long ans = 0;

        foreach (var kv in segments) {
            ans += Count(kv.Value);
        }
        //segments.Dump();
        return ans;
    }

    static private long Count(List<Tuple<long, long>> segments) {
        segments.Sort((v1, v2) => v1.Item1.CompareTo(v2.Item1));
        long left = long.MinValue;
        long right = long.MinValue;
        long ans = 0;
        foreach (var seg in segments) {
            if (seg.Item1 <= right) { // have common
                right = Math.Max(right, seg.Item2);
            } else {
                if (left != long.MinValue) {
                    ans += right - left + 1;
                }
                left = seg.Item1;
                right = seg.Item2;
            }
        }
        if (left != long.MinValue) {
            ans += right - left + 1;
        }
        return ans;
    }

    public static long DivLower(long x, long y) {
        if (y < 0) {
            return DivLower(-x, -y);
        }

        if (x >= 0) {
            return x / y;
        }

        return x / y - (x % y == 0 ? 0 : 1);
    }
}

// PREWRITEN CODE BEGINS FROM HERE
internal partial class Solver : Scanner {
    public static void Main(string[] args) {
#if LOCAL
        byte[] inputBuffer = new byte[1000000];
        var inputStream = Console.OpenStandardInput(inputBuffer.Length);
        using (var reader = new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length)) {
            Console.SetIn(reader);
            new Solver(Console.In, Console.Out).Run();
        }
#else
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) { AutoFlush = false });
        new Solver(Console.In, Console.Out).Run();
        Console.Out.Flush();
#endif
    }

#pragma warning disable IDE0052
    private readonly TextReader cin;
    private readonly TextWriter cout;
#pragma warning restore IDE0052

    public Solver(TextReader reader, TextWriter writer)
        : base(reader) {
        cin = reader;
        cout = writer;
    }
    public Solver(string input, TextWriter writer)
        : this(new StringReader(input), writer) {
    }

#pragma warning disable IDE1006
#pragma warning disable IDE0051
    private int ni() { return NextInt(); }
    private int[] ni(int n) { return NextIntArray(n); }
    private long nl() { return NextLong(); }
    private long[] nl(int n) { return NextLongArray(n); }
    private double nd() { return NextDouble(); }
    private double[] nd(int n) { return NextDoubleArray(n); }
    private string ns() { return Next(); }
    private string[] ns(int n) { return NextArray(n); }
#pragma warning restore IDE1006
#pragma warning restore IDE0051
}

internal static class LinqPadExtension {
    public static T Dump<T>(this T obj) {
#if LOCAL
        return LINQPad.Extensions.Dump(obj);
#else
        return obj;
#endif
    }
}

public class Scanner {
    private readonly TextReader Reader;
    private readonly Queue<string> TokenQueue = new Queue<string>();
    private readonly CultureInfo ci = CultureInfo.InvariantCulture;

    public Scanner()
        : this(Console.In) {
    }

    public Scanner(TextReader reader) {
        Reader = reader;
    }

    public int NextInt() { return int.Parse(Next(), ci); }
    public long NextLong() { return long.Parse(Next(), ci); }
    public double NextDouble() { return double.Parse(Next(), ci); }
    public string[] NextArray(int size) {
        string[] array = new string[size];
        for (int i = 0; i < size; i++) {
            array[i] = Next();
        }

        return array;
    }
    public int[] NextIntArray(int size) {
        int[] array = new int[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextInt();
        }

        return array;
    }

    public long[] NextLongArray(int size) {
        long[] array = new long[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextLong();
        }

        return array;
    }

    public double[] NextDoubleArray(int size) {
        double[] array = new double[size];
        for (int i = 0; i < size; i++) {
            array[i] = NextDouble();
        }

        return array;
    }

    public string Next() {
        if (TokenQueue.Count == 0) {
            if (!StockTokens()) {
                throw new InvalidOperationException();
            }
        }
        return TokenQueue.Dequeue();
    }

    public bool HasNext() {
        if (TokenQueue.Count > 0) {
            return true;
        }

        return StockTokens();
    }

    private static readonly char[] _separator = new[] { ' ', '\t' };
    private bool StockTokens() {
        while (true) {
            string line = Reader.ReadLine();
            if (line == null) {
                return false;
            }

            string[] tokens = line.Split(_separator, StringSplitOptions.RemoveEmptyEntries);
            if (tokens.Length == 0) {
                continue;
            }

            foreach (string token in tokens) {
                TokenQueue.Enqueue(token);
            }

            return true;
        }
    }
}