Sum of Vectors

#include <bits/stdc++.h>

using namespace std;

using Vector = pair<int, int>;
#define x first
#define y second

int cross(Vector x, Vector y) {
    return x.x * y.y - x.y * y.x;
}

int getQuater(Vector vec) {
    auto [x, y] = vec;
    if (x >= 0 && y >= 0) return 0;
    if (x < 0 && y >= 0) return 1;
    if (x < 0 && y < 0) return 2;
    if (x >= 0 && y < 0) return 3;
    return -1;
}

bool comp(Vector vec1, Vector vec2) {
    int quater1 = getQuater(vec1);
    int quater2 = getQuater(vec2);
    if (quater1 != quater2) return quater1 < quater2;
    return cross(vec1, vec2) > 0;
}

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    int n;
    cin >> n;
    vector<Vector> vecs(n);
    for (auto &vec : vecs) cin >> vec.x >> vec.y;
    sort(vecs.begin(), vecs.end(), comp);

    long long best = 0;
    int sumX = 0, sumY = 0;

    int p = n - 1, cnt = 0;

    auto update = [&]() {
        best = max(best, 1LL * sumX * sumX + 1LL * sumY * sumY);
    };

    for (int i = 0; i < n; i++) {
        while (cnt < n && cross(vecs[i], vecs[(p + 1) % n]) >= 0) {
            p = (p + 1) % n;
            sumX += vecs[p].x, sumY += vecs[p].y;
            cnt++;
            update();
        }
        cnt--;
        sumX -= vecs[i].x, sumY -= vecs[i].y;
        update();
    }

    cout << best << '\n';

    return 0;
}