Problem:
Score : 400 points

Problem Statement

Takahashi has N sticks that are distinguishable from each other. The length of the i-th stick is L_i.

He is going to form a triangle using three of these sticks. Let a, b, and c be the lengths of the three sticks used. Here, all of the following conditions must be satisfied:

a < b + c

b < c + a

c < a + b

How many different triangles can be formed? Two triangles are considered different when there is a stick used in only one of them.

Constraints

All values in input are integers.

3 \leq N \leq 2 \times 10^3

1 \leq L_i \leq 10^3

Input

Input is given from Standard Input in the following format:

N
L_1 L_2 ... L_N

Constraints

Print the number of different triangles that can be formed.

Sample Input 1

4
3 4 2 1

Sample Output 1

1

Only one triangle can be formed: the triangle formed by the first, second, and third sticks.

Sample Input 2

3
1 1000 1

Sample Output 2

0

No triangles can be formed.

Sample Input 3

7
218 786 704 233 645 728 389

Sample Output 3

23

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.*;
import java.awt.geom.Point2D;
import java.io.IOException;
import java.io.InputStream;
import java.util.*;
import java.util.stream.*;
import java.util.function.BiFunction;
import java.util.function.Function;
import java.util.function.Supplier;

class Main {
    final static int INF = 1 << 28;
    final static long LNF = 1L << 60;
    final static double EPS = 1e-9;
    final static double GOLDEN_RATIO = (1.0 + Math.sqrt(5)) / 2.0;

    static final long MOD = 1_000_000_007;

    public static void main(String[] args) throws Exception {
        final MyScanner sc = new MyScanner();
        long N;
        N = sc.nextLong();
        long[] L = new long[(int)(N)];
        for(int i = 0 ; i < N ; i++){
                L[i] = sc.nextLong();
        }
        solve(N, L);
    }

    static void solve(long N, long[] L){
        Arrays.sort(L);
        HashMap<Integer, Integer> hmap = new HashMap<Integer, Integer>();
        long hoge = 0;
        for (int i = 0; i < L.length; i++) {
            if(hoge != L[i]) {
                hmap.put((int)L[i], i); 
                hoge = L[i];
            }
        }
        

        long ans = 0;
        for (long i = 0; i < L.length - 2; i++) {
            for (long j = i + 1; j < L.length - 1; j++) {
                long idx = Arrays.binarySearch(L, (int) (j + 1), L.length, L[(int) i] + L[(int) j]);
                if(idx<0) idx = -(idx+1);
                else idx = hmap.get((int)(L[(int) i] + L[(int) j]));
                ans += idx - (j+1);
            }
        }
        System.out.println(ans);
    }




    /**
     * ユークリッドの互除法
     *
     * @return a と b の最大公約数
     */
    static long gcd(long a, long b) {
        if (b == 0) {
            return a;
        }
        return gcd(b, a % b);
    }

    /**
     * 拡張ユークリッドの互除法
     *
     * @return mx + ny = gcd(m, n)となるような(x, y)を返す
     */
    static Pair<Long, Long> gcd_ex(long m, long n) {
        long[][] mat = _gcd_ex(m, n);
        return new Pair<>(mat[0][0], mat[0][1]);
    }

    static long[][] _gcd_ex(long m, long n) {
        if (n == 0) {
            return new long[][] { { 1, 0 }, { 0, 1 } };
        }
        long k = m / n;
        long[][] K = new long[][] { { 0, 1 }, { 1, -k } };
        long[][] r = _gcd_ex(n, m % n);
        long[][] dst = new long[2][2];
        for (int y = 0; y < 2; ++y)
            for (int x = 0; x < 2; ++x)
                for (int i = 0; i < 2; ++i)
                    dst[y][x] += r[y][i] * K[i][x];
        return dst;
    }

    /**
     * 繰り返し2乗法を用いたべき乗の実装
     *
     * @return a^r (mod 1,000,000,007)
     */
    static long pow(long a, long r) {
        long sum = 1;
        while (r > 0) {
            if ((r & 1) == 1) {
                sum *= a;
                sum %= MOD;
            }
            a *= a;
            a %= MOD;
            r >>= 1;
        }
        return sum;
    }

    /**
     * 組み合わせ O(n)
     *
     * @return {}_nC_r
     */
    static long C(int n, int r) {
        long sum = 1;
        for (int i = n; 0 < i; --i) {
            sum *= i;
            sum %= MOD;
        }
        long s = 1;
        for (int i = r; 0 < i; --i) {
            s *= i;
            s %= MOD;
        }
        sum *= pow(s, MOD - 2);
        sum %= MOD;

        long t = 1;
        for (int i = n - r; 0 < i; --i) {
            t *= i;
            t %= MOD;
        }
        sum *= pow(t, MOD - 2);
        sum %= MOD;

        return sum;
    }

    /**
     * 黄金分割探索
     *
     * @param left  下限
     * @param right 上限
     * @param f     探索する関数
     * @param comp  上に凸な関数を探索するときは、Comparator.comparingDouble(Double::doubleValue)
     *              下に凸な関数を探索するときは、Comparator.comparingDouble(Double::doubleValue).reversed()
     * @return 極値の座標x
     */
    static double goldenSectionSearch(double left, double right, Function<Double, Double> f, Comparator<Double> comp) {
        double c1 = divideInternally(left, right, 1, GOLDEN_RATIO);
        double c2 = divideInternally(left, right, GOLDEN_RATIO, 1);
        double d1 = f.apply(c1);
        double d2 = f.apply(c2);
        while (right - left > 1e-9) {
            if (comp.compare(d1, d2) > 0) {
                right = c2;
                c2 = c1;
                d2 = d1;
                c1 = divideInternally(left, right, 1, GOLDEN_RATIO);
                d1 = f.apply(c1);
            } else {
                left = c1;
                c1 = c2;
                d1 = d2;
                c2 = divideInternally(left, right, GOLDEN_RATIO, 1);
                d2 = f.apply(c2);
            }
        }
        return right;
    }

    /**
     * [a,b]をm:nに内分する点を返す
     */
    static double divideInternally(double a, double b, double m, double n) {
        return (n * a + m * b) / (m + n);
    }

    /**
     * http://alexbowe.com/popcount-permutations/ bitの立っている数が小さい順にループしたいときに使う。 ex)
     * 
     * <pre>
     * for (int i = 0; i < 25; ++i) {
     *   int bits = (1 << i) - 1;
     *   long m = C(25, num);
     *   for (j = 0; j < m; ++j) {
     *     ...(25個の中からi個bitが立っている)
     *     if (bits != 0)
     *       bits = next_perm(bits);
     *   }
     * }
     * </pre>
     *
     * @param v 現在のbit列
     * @return 次のbit列
     */
    static int next_perm(int v) {
        int t = (v | (v - 1)) + 1;
        return t | ((((t & -t) / (v & -v)) >> 1) - 1);
    }

    static class Util {
        static <T extends Comparable<T>> T max(T a, T b) {
            if (a.compareTo(b) > 0) {
                return a;
            } else {
                return b;
            }
        }

        static <T extends Comparable<T>> T min(T a, T b) {
            if (a.compareTo(b) < 0) {
                return a;
            } else {
                return b;
            }
        }
    }

    /**
     * from http://gihyo.jp/dev/serial/01/geometry part 6
     */
    static class Line {
        double a;
        double b;
        double c;

        /**
         * 一般形のパラメータから直線を作成する
         *
         * @param a xの係数
         * @param b yの係数
         * @param c 定数項
         */
        Line(double a, double b, double c) {
            this.a = a;
            this.b = b;
            this.c = c;
        }

        /**
         * 2点(x1, y1), (x2, y2)を通る直線を作成する
         *
         * @param x1 1点目のx座標
         * @param y1 1点目のy座標
         * @param x2 2点目のx座標
         * @param y2 2点目のy座標
         * @return 直線
         */
        static Line fromPoints(double x1, double y1, double x2, double y2) {
            double dx = x2 - x1;
            double dy = y2 - y1;
            return new Line(dy, -dx, dx * y1 - dy * x1);
        }

        /**
         * 与えられた直線との交点を返す
         *
         * @param l 直線
         * @return 交点。2直線が平行の場合はnull
         */
        Point2D getIntersectionPoint(Line l) {
            double d = a * l.b - l.a * b;
            if (d == 0.0) {
                return null;
            }
            double x = (b * l.c - l.b * c) / d;
            double y = (l.a * c - a * l.c) / d;
            return new Point2D.Double(x, y);
        }

        @Override
        public String toString() {
            return "a = " + a + ", b = " + b + ", c = " + c;
        }
    }

    static class Line1D {
        int x1;
        int x2;

        // [x1, x2) : 半開空間
        Line1D(int x1, int x2) {
            this.x1 = x1;
            this.x2 = x2;
        }

        boolean isCross(Line1D l) {
            return isCross(l.x1, l.x2);
        }

        boolean isCross(int y1, int y2) {
            boolean ret = x1 < y2 && y1 < x2;
            assert ret == new LineSegment(x1, 0, x2, 0).intersects(new LineSegment(y1, 0, y2, 0));
            return ret;
        }
    }

    /**
     * from http://gihyo.jp/dev/serial/01/geometry part 6
     */
    static public class LineSegment {
        double x1;
        double y1;
        double x2;
        double y2;

        LineSegment(double x1, double y1, double x2, double y2) {
            this.x1 = x1;
            this.y1 = y1;
            this.x2 = x2;
            this.y2 = y2;
        }

        Line toLine() {
            return Line.fromPoints(x1, y1, x2, y2);
        }

        boolean intersects(Line l) {
            double t1 = l.a * x1 + l.b * y1 + l.c;
            double t2 = l.a * x2 + l.b * y2 + l.c;
            return t1 * t2 <= 0;
        }

        boolean intersects(LineSegment s) {
            return bothSides(s) && s.bothSides(this);
        }

        // sが自線分の「両側」にあるかどうかを調べる
        private boolean bothSides(LineSegment s) {
            double ccw1 = GeomUtils.ccw(x1, y1, s.x1, s.y1, x2, y2);
            double ccw2 = GeomUtils.ccw(x1, y1, s.x2, s.y2, x2, y2);
            if (ccw1 == 0 && ccw2 == 0) { // sと自線分が一直線上にある場合
                // sのいずれか1つの端点が自線分を内分していれば、sは自線分と共有部分を持つので
                // trueを返す
                return internal(s.x1, s.y1) || internal(s.x2, s.y2);
            } else { // それ以外の場合
                // CCW値の符号が異なる場合にtrueを返す
                return ccw1 * ccw2 <= 0;
            }
        }

        // (x, y)が自線分を内分しているかどうかを調べる
        private boolean internal(double x, double y) {
            // (x, y)から端点に向かうベクトルの内積がゼロ以下であれば内分と見なす
            return GeomUtils.dot(x1 - x, y1 - y, x2 - x, y2 - y) <= 0;
        }

        public Point2D getIntersectionPoint(Line l) {
            if (!intersects(l)) {
                return null; // 交差しない場合はnullを返す
            }
            return l.getIntersectionPoint(toLine());
        }

        public Point2D getIntersectionPoint(LineSegment s) {
            if (!intersects(s)) {
                return null; // 交差しない場合はnullを返す
            }
            return s.toLine().getIntersectionPoint(toLine());
        }

        /**
         * from :
         * http://www.deqnotes.net/acmicpc/2d_geometry/lines#distance_between_line_segment_and_point
         */
        double distance(double x0, double y0) {
            // 端点チェック
            if (GeomUtils.dot(x2 - x1, y2 - y1, x0 - x1, y0 - y1) < EPS) {
                return GeomUtils.abs(x0 - x1, y0 - y1);
            }
            if (GeomUtils.dot(x1 - x2, y1 - y2, x0 - x2, y0 - y2) < EPS) {
                return GeomUtils.abs(x0 - x2, y0 - y2);
            }
            // 直線と点の距離
            return Math.abs(GeomUtils.cross(x2 - x1, y2 - y1, x0 - x1, y0 - y1)) / GeomUtils.abs(x2 - x1, y2 - y1);
        }

        double distance(LineSegment l) {
            if (this.intersects(l)) {
                return 0.0;
            }
            double min = Double.MAX_VALUE;
            min = Math.min(min, distance(l.x1, l.y1));
            min = Math.min(min, distance(l.x2, l.y2));
            min = Math.min(min, l.distance(x1, y1));
            min = Math.min(min, l.distance(x2, y2));
            return min;
        }

        @Override
        public String toString() {
            return "(" + x1 + ", " + y1 + ") - (" + x2 + ", " + y2 + ")";
        }
    }

    /**
     * from http://gihyo.jp/dev/serial/01/geometry part 6
     */
    static class GeomUtils {
        // 外積
        static double cross(double x1, double y1, double x2, double y2) {
            return x1 * y2 - x2 * y1;
        }

        // 内積
        static double dot(double x1, double y1, double x2, double y2) {
            return x1 * x2 + y1 * y2;
        }

        // (x1, y1) -> (x2, y2) -> (x3, y3) と進む道のりが半時計回りの場合は正の値、
        // 時計回りの場合は負の値、一直線上の場合はゼロを返す
        static double ccw(double x1, double y1, double x2, double y2, double x3, double y3) {
            return cross(x2 - x1, y2 - y1, x3 - x2, y3 - y2);
        }

        static double ccw(Point2D p1, Point2D p2, Point2D p3) {
            return ccw(p1.getX(), p1.getY(), p2.getX(), p2.getY(), p3.getX(), p3.getY());
        }

        // ベクトルの長さ
        static double abs(double x, double y) {
            return Math.sqrt(x * x + y * y);
        }

        // 多角形の内外判定
        // http://www.nttpc.co.jp/technology/number_algorithm.html
        // 模擬国内2017EでWAになった
        // 凸多角形 -> 右周りでccwが全部負
        // 凹多角形含む単純な多角形 -> 点から無限に右に伸ばした線分との交点が1つ
        static boolean isIn(ArrayList<LineSegment> list, double x, double y) {
            int wn = 0;
            for (LineSegment l : list) {
                if (l.y1 <= y && l.y2 > y) {
                    double vt = (y - l.y1) / (l.y2 - l.y1);
                    if (x < l.x1 + (vt * (l.x2 - l.x1))) {
                        ++wn;
                    }
                }
                if (l.y1 > y && l.y2 <= y) {
                    double vt = (y - l.y1) / (l.y2 - l.y1);
                    if (x < (l.x1 + (vt * (l.x2 - l.x1)))) {
                        --wn;
                    }
                }
            }
            return wn > 0;
        }
    }

    static class MyScanner {
 
        BufferedReader br;
        StringTokenizer st;
 
        MyScanner(FileReader fileReader) {
            br = new BufferedReader(fileReader);
        }
 
        MyScanner() {
            br = new BufferedReader(new InputStreamReader(System.in));
        }
 
        String nn() {
            while (st == null || !st.hasMoreElements()) {
                try {
                    st = new StringTokenizer(br.readLine());
                } catch (IOException e) {
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }
 
        String next() {
            String ans = "";
            try {
                ans = br.readLine();
            } catch (IOException ex) {
                ex.printStackTrace();
            }
            return ans;
        }
 
        char nextChar() {
            return nn().charAt(0);
        }
 
        int nextInt() {
            return Integer.parseInt(nn());
        }
 
        long nextLong() {
            return Long.parseLong(nn());
        }
 
        double nextDouble() {
            return Double.parseDouble(nn());
        }
 
        Integer[] niArr0(int n) {
            Integer[] ar = new Integer[n];
            for (int i = 0; i < n; i++) {
                ar[i] = nextInt();
            }
            return ar;
        }
 
        Integer[] niArr1(int n) {
            Integer[] ar = new Integer[n + 1];
            for (int i = 1; i <= n; i++) {
                ar[i] = nextInt();
            }
            return ar;
        }
 
        Long[] nlArr0(int n) {
            Long[] ar = new Long[n];
            for (int i = 0; i < n; i++) {
                ar[i] = nextLong();
            }
            return ar;
        }
 
        Long[] nlArr1(int n) {
            Long[] ar = new Long[n + 1];
            for (int i = 1; i <= n; i++) {
                ar[i] = nextLong();
            }
            return ar;
        }
    }

    static class Pair<F extends Comparable<F>, S extends Comparable<S>> implements Comparable<Pair<F, S>> {
        F f;
        S s;

        Pair() {
        }

        Pair(F f, S s) {
            this.f = f;
            this.s = s;
        }

        Pair(Pair<F, S> p) {
            f = p.f;
            s = p.s;
        }

        @Override
        public int compareTo(Pair<F, S> p) {
            if (f.compareTo(p.f) != 0) {
                return f.compareTo(p.f);
            }
            return s.compareTo(p.s);
        }

        @Override
        public int hashCode() {
            return f.hashCode() ^ s.hashCode();
        }

        @Override
        public boolean equals(Object o) {
            if (this == o) {
                return true;
            }
            if (o == null || this.f == null || this.s == null) {
                return false;
            }
            if (this.getClass() != o.getClass()) {
                return false;
            }
            Pair p = (Pair) o;
            return this.f.equals(p.f) && this.s.equals(p.s);
        }

        @Override
        public String toString() {
            return "{" + f.toString() + ", " + s.toString() + "}";
        }
    }

    class BIT<T> {
        int n;
        ArrayList<T> bit;
        BiFunction<T, T, T> bif;

        /**
         * 1-indexed なBinary Indexed Treeを構築する
         *
         * @param n   容量
         * @param bif 適用させる関数
         * @param sup 初期値
         */
        BIT(int n, BiFunction<T, T, T> bif, Supplier<T> sup) {
            this.n = n;
            bit = new ArrayList<>(n + 1);
            for (int i = 0; i < n + 1; ++i) {
                bit.add(sup.get());
            }
            this.bif = bif;
        }

        /**
         * iの位置の値をvで更新する
         *
         * @param i index
         * @param v 新しい値
         */
        void set(int i, T v) {
            for (int x = i; x <= n; x += x & -x) {
                bit.set(x, bif.apply(bit.get(x), v));
            }
        }

        /**
         * クエリー
         *
         * @param defaultValue 初期値
         * @param i            index
         * @return [1, i]までfを適用した結果
         */
        T reduce(T defaultValue, int i) {
            T ret = defaultValue;
            for (int x = i; x > 0; x -= x & -x) {
                ret = bif.apply(ret, bit.get(x));
            }
            return ret;
        }
    }

    class SegmentTree<T> {
        int n;
        ArrayList<T> dat;
        BiFunction<T, T, T> bif;
        Supplier<T> sup;

        /**
         * 0-indexed なSegment Treeを構築する
         *
         * @param n_  要求容量
         * @param bif 適用させる関数
         * @param sup 初期値
         */
        SegmentTree(int n_, BiFunction<T, T, T> bif, Supplier<T> sup) {
            n = 1;
            while (n < n_)
                n *= 2;

            dat = new ArrayList<>(2 * n - 1);
            for (int i = 0; i < 2 * n - 1; ++i) {
                dat.add(sup.get());
            }
            this.bif = bif;
            this.sup = sup;
        }

        /**
         * kの位置の値をvで更新する
         *
         * @param k index
         * @param v 新しい値
         */
        void set(int k, T v) {
            k += n - 1;
            dat.set(k, v);
            while (k > 0) {
                k = (k - 1) / 2;
                dat.set(k, bif.apply(dat.get(k * 2 + 1), dat.get(k * 2 + 2)));
            }
        }

        /**
         * クエリー
         *
         * @param l はじめ
         * @param r おわり
         * @return [l, r)での演算bifを適用した結果を返す
         */
        T reduce(int l, int r) {
            return _reduce(l, r, 0, 0, n);
        }

        T _reduce(int a, int b, int k, int l, int r) {
            if (r <= a || b <= l)
                return sup.get();
            if (a <= l && r <= b)
                return dat.get(k);
            T vl = _reduce(a, b, k * 2 + 1, l, (l + r) / 2);
            T vr = _reduce(a, b, k * 2 + 2, (l + r) / 2, r);
            return bif.apply(vl, vr);
        }
    }

    class UnionFind {
        int[] par;

        UnionFind(int n) {
            par = new int[n];
            for (int i = 0; i < n; ++i) {
                par[i] = i;
            }
        }

        int find(int x) {
            if (par[x] == x) {
                return x;
            }
            return par[x] = find(par[x]);
        }

        boolean same(int x, int y) {
            return find(x) == find(y);
        }

        void union(int x, int y) {
            x = find(x);
            y = find(y);
            if (x == y) {
                return;
            }
            par[x] = y;
        }
    }
}