Problem:
Score : 600 points

Problem Statement

Given are a prime number p and a sequence of p integers a_0, \ldots, a_{p-1} consisting of zeros and ones.

Find a polynomial of degree at most p-1, f(x) = b_{p-1} x^{p-1} + b_{p-2} x^{p-2} + \ldots + b_0, satisfying the following conditions:

For each i (0 \leq i \leq p-1), b_i is an integer such that 0 \leq b_i \leq p-1.

For each i (0 \leq i \leq p-1), f(i) \equiv a_i \pmod p.

Constraints

2 \leq p \leq 2999

p is a prime number.

0 \leq a_i \leq 1

Input

Input is given from Standard Input in the following format:

p
a_0 a_1 \ldots a_{p-1}

Output

Print b_0, b_1, \ldots, b_{p-1} of a polynomial f(x) satisfying the conditions, in this order, with spaces in between.

It can be proved that a solution always exists. If multiple solutions exist, any of them will be accepted.

Sample Input 1

2
1 0

Sample Output 1

1 1

f(x) = x + 1 satisfies the conditions, as follows:

f(0) = 0 + 1 = 1 \equiv 1 \pmod 2

f(1) = 1 + 1 = 2 \equiv 0 \pmod 2

Sample Input 2

3
0 0 0

Sample Output 2

0 0 0

f(x) = 0 is also valid.

Sample Input 3

5
0 1 0 1 0

Sample Output 3

0 2 0 1 3

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.Linq;

using E = System.Linq.Enumerable;

internal partial class Solver {
    public void Run() {
        var P = ni();
        var a = nl(P);

        var x = E.Range(0, P).Select(i => (long)i).ToArray();
        var ans = LagrangePolynomial.Build(x, a, P);
        cout.WriteLine(string.Join(" ", ans.GetCoeffients()));
    }

}

public class PolynomialMod {
    private readonly long[] _coefficients;
    private readonly int Mod;

    public PolynomialMod(long[] coefficient, int mod) {
        _coefficients = coefficient.ToArray();
        Mod = mod;
    }

    public long[] GetCoeffients() {
        return _coefficients.ToArray();
    }

    public long Invoke(long x) {
        x = (x % Mod + Mod) % Mod;
        long t = 1;
        long result = 0;
        foreach (var c in _coefficients) {
            result += t * c;
            result %= Mod;
            t *= x;
            t %= Mod;
        }
        return result;
    }
}

public static class LagrangePolynomial {
    private static long ExtendedGcd(long a, long b, ref long x, ref long y) {
        if (b == 0) { x = 1; y = 0; return a; } else {
            long d = ExtendedGcd(b, a % b, ref y, ref x);
            y -= a / b * x;
            return d;
        }
    }
    private static long Inverse(long a, long mod) {
        long x = 0, y = 0;
        if (ExtendedGcd(a, mod, ref x, ref y) == 1) {
            return (x + mod) % mod;
        } else {
            return -1;
        }
    }

    private static long Normalize(long value, int mod) {
        value %= mod;
        if (value < 0) value += mod;
        return value;
    }

    public static PolynomialMod Build(long[] x, long[] y, int mod) {
        if (x == null) {
            throw new ArgumentNullException("x");
        }

        if (y == null) {
            throw new ArgumentNullException("y");
        }

        if (x.Length != y.Length) {
            throw new InvalidOperationException();
        }

        int N = x.Length;
        long[] X = new long[N];
        long[] Y = new long[N];
        for (int i = 0; i < N; i++) {
            X[i] = Normalize(x[i], mod);
            Y[i] = Normalize(y[i], mod);
        }

        long[] allPoly = new long[N + 1];
        // (x - x1)(x - x2) .. (x - xn)
        allPoly[0] = 1;
        for (int i = 0; i < N; i++) {
            for (int j = N; j >= 0; j--) {
                allPoly[j] = (-X[i] * allPoly[j] + (j - 1 >= 0 ? allPoly[j - 1] : 0)) % mod;
            }
        }
        for (int i = 0; i < N; i++) {
            allPoly[i] = Normalize(allPoly[i], mod);
        }

        long[] c = new long[N];
        long[] poly = new long[N + 1];

        for (int i = 0; i < N; i++) {
            long invTerm = 1;
            for (int j = 0; j < N; j++) {
                if (i != j) {
                    invTerm = invTerm * ((X[i] - X[j] + mod) % mod) % mod;
                }
            }
            long coef = Inverse(invTerm, mod) * Y[i] % mod;

            // (x - x1)(x - x2) .. (x - xn) / (x - xi)
            Array.Copy(allPoly, poly, allPoly.Length);
            for (int j = N; j >= 1; j--) {
                long t = poly[j];
                poly[j - 1] = (poly[j - 1] - t * -X[i]) % mod;
                c[j - 1] = (c[j - 1] + coef * t) % mod;
            }
        }

        for (int i = 0; i < N; i++) {
            c[i] = Normalize(c[i], mod);
        }

        return new PolynomialMod(c, mod);
    }
}


// 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);
        Console.SetIn(new StreamReader(inputStream, Console.InputEncoding, false, inputBuffer.Length));
        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;
        }
    }
}