結果

問題 No.2262 Fractions
ユーザー tokusakurai
提出日時 2023-04-07 23:16:40
言語 C++17(gcc12)
(gcc 12.3.0 + boost 1.87.0)
結果
AC  
実行時間 1,419 ms / 2,000 ms
コード長 12,483 bytes
コンパイル時間 10,496 ms
コンパイル使用メモリ 290,528 KB
最終ジャッジ日時 2025-02-12 02:40:19
ジャッジサーバーID
（参考情報） 		judge4 / judge5
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ファイルパターン 結果
sample AC * 1
other AC * 45
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ソースコード

diff #
プレゼンテーションモードにする

#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
#define rep(i, n) for (int i = 0; i < (n); i++)
#define per(i, n) for (int i = (n)-1; i >= 0; i--)
#define rep2(i, l, r) for (int i = (l); i < (r); i++)
#define per2(i, l, r) for (int i = (r)-1; i >= (l); i--)
#define each(e, v) for (auto &e : v)
#define MM << " " <<
#define pb push_back
#define eb emplace_back
#define all(x) begin(x), end(x)
#define rall(x) rbegin(x), rend(x)
#define sz(x) (int)x.size()
using ll = long long;
using pii = pair<int, int>;
using pil = pair<int, ll>;
using pli = pair<ll, int>;
using pll = pair<ll, ll>;
template <typename T>
using minheap = priority_queue<T, vector<T>, greater<T>>;
template <typename T>
using maxheap = priority_queue<T>;
template <typename T>
bool chmax(T &x, const T &y) {
    return (x < y) ? (x = y, true) : false;
}
template <typename T>
bool chmin(T &x, const T &y) {
    return (x > y) ? (x = y, true) : false;
}
template <typename T>
int flg(T x, int i) {
    return (x >> i) & 1;
}
int popcount(int x) { return __builtin_popcount(x); }
int popcount(ll x) { return __builtin_popcountll(x); }
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int botbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int botbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
template <typename T>
void print(const vector<T> &v, T x = 0) {
    int n = v.size();
    for (int i = 0; i < n; i++) cout << v[i] + x << (i == n - 1 ? '\n' : ' ');
    if (v.empty()) cout << '\n';
}
template <typename T>
void printn(const vector<T> &v, T x = 0) {
    int n = v.size();
    for (int i = 0; i < n; i++) cout << v[i] + x << '\n';
}
template <typename T>
int lb(const vector<T> &v, T x) {
    return lower_bound(begin(v), end(v), x) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, T x) {
    return upper_bound(begin(v), end(v), x) - begin(v);
}
template <typename T>
void rearrange(vector<T> &v) {
    sort(begin(v), end(v));
    v.erase(unique(begin(v), end(v)), end(v));
}
template <typename T>
vector<int> id_sort(const vector<T> &v, bool greater = false) {
    int n = v.size();
    vector<int> ret(n);
    iota(begin(ret), end(ret), 0);
    sort(begin(ret), end(ret), [&](int i, int j) { return greater ? v[i] > v[j] : v[i] < v[j]; });
    return ret;
}
template <typename T>
void reorder(vector<T> &a, const vector<int> &ord) {
    int n = a.size();
    vector<T> b(n);
    for (int i = 0; i < n; i++) b[i] = a[ord[i]];
    swap(a, b);
}
template <typename T>
T floor(T x, T y) {
    assert(y != 0);
    if (y < 0) x = -x, y = -y;
    return (x >= 0 ? x / y : (x - y + 1) / y);
}
template <typename T>
T ceil(T x, T y) {
    assert(y != 0);
    if (y < 0) x = -x, y = -y;
    return (x >= 0 ? (x + y - 1) / y : x / y);
}
template <typename S, typename T>
pair<S, T> operator+(const pair<S, T> &p, const pair<S, T> &q) {
    return make_pair(p.first + q.first, p.second + q.second);
}
template <typename S, typename T>
pair<S, T> operator-(const pair<S, T> &p, const pair<S, T> &q) {
    return make_pair(p.first - q.first, p.second - q.second);
}
template <typename S, typename T>
istream &operator>>(istream &is, pair<S, T> &p) {
    S a;
    T b;
    is >> a >> b;
    p = make_pair(a, b);
    return is;
}
template <typename S, typename T>
ostream &operator<<(ostream &os, const pair<S, T> &p) {
    return os << p.first << ' ' << p.second;
}
struct io_setup {
    io_setup() {
        ios_base::sync_with_stdio(false);
        cin.tie(NULL);
        cout << fixed << setprecision(15);
    }
} io_setup;
const int inf = (1 << 30) - 1;
const ll INF = (1LL << 60) - 1;
// const int MOD = 1000000007;
const int MOD = 998244353;
struct Hopcroft_Karp {
    vector<vector<int>> es;
    vector<int> d, match;
    vector<bool> used, used2;
    const int n, m;
    Hopcroft_Karp(int n, int m) : es(n), d(n), match(m), used(n), used2(n), n(n), m(m) {}
    void add_edge(int u, int v) { es[u].push_back(v); }
    void _bfs() {
        fill(begin(d), end(d), -1);
        queue<int> que;
        for (int i = 0; i < n; i++) {
            if (!used[i]) {
                que.push(i);
                d[i] = 0;
            }
        }
        while (!que.empty()) {
            int i = que.front();
            que.pop();
            for (auto &e : es[i]) {
                int j = match[e];
                if (j != -1 && d[j] == -1) {
                    que.push(j);
                    d[j] = d[i] + 1;
                }
            }
        }
    }
    bool _dfs(int now) {
        used2[now] = true;
        for (auto &e : es[now]) {
            int u = match[e];
            if (u == -1 || (!used2[u] && d[u] == d[now] + 1 && _dfs(u))) {
                match[e] = now, used[now] = true;
                return true;
            }
        }
        return false;
    }
    int max_matching() { // 右側の i は左側の match[i] とマッチングする
        fill(begin(match), end(match), -1), fill(begin(used), end(used), false);
        int ret = 0;
        while (true) {
            _bfs();
            fill(begin(used2), end(used2), false);
            int flow = 0;
            for (int i = 0; i < n; i++) {
                if (!used[i] && _dfs(i)) flow++;
            }
            if (flow == 0) break;
            ret += flow;
        }
        return ret;
    }
};
template <int mod>
struct Mod_Int {
    int x;
    Mod_Int() : x(0) {}
    Mod_Int(long long y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}
    static int get_mod() { return mod; }
    Mod_Int &operator+=(const Mod_Int &p) {
        if ((x += p.x) >= mod) x -= mod;
        return *this;
    }
    Mod_Int &operator-=(const Mod_Int &p) {
        if ((x += mod - p.x) >= mod) x -= mod;
        return *this;
    }
    Mod_Int &operator*=(const Mod_Int &p) {
        x = (int)(1LL * x * p.x % mod);
        return *this;
    }
    Mod_Int &operator/=(const Mod_Int &p) {
        *this *= p.inverse();
        return *this;
    }
    Mod_Int &operator++() { return *this += Mod_Int(1); }
    Mod_Int operator++(int) {
        Mod_Int tmp = *this;
        ++*this;
        return tmp;
    }
    Mod_Int &operator--() { return *this -= Mod_Int(1); }
    Mod_Int operator--(int) {
        Mod_Int tmp = *this;
        --*this;
        return tmp;
    }
    Mod_Int operator-() const { return Mod_Int(-x); }
    Mod_Int operator+(const Mod_Int &p) const { return Mod_Int(*this) += p; }
    Mod_Int operator-(const Mod_Int &p) const { return Mod_Int(*this) -= p; }
    Mod_Int operator*(const Mod_Int &p) const { return Mod_Int(*this) *= p; }
    Mod_Int operator/(const Mod_Int &p) const { return Mod_Int(*this) /= p; }
    bool operator==(const Mod_Int &p) const { return x == p.x; }
    bool operator!=(const Mod_Int &p) const { return x != p.x; }
    Mod_Int inverse() const {
        assert(*this != Mod_Int(0));
        return pow(mod - 2);
    }
    Mod_Int pow(long long k) const {
        Mod_Int now = *this, ret = 1;
        for (; k > 0; k >>= 1, now *= now) {
            if (k & 1) ret *= now;
        }
        return ret;
    }
    friend ostream &operator<<(ostream &os, const Mod_Int &p) { return os << p.x; }
    friend istream &operator>>(istream &is, Mod_Int &p) {
        long long a;
        is >> a;
        p = Mod_Int<mod>(a);
        return is;
    }
};
using mint = Mod_Int<MOD>;
template <typename T>
vector<T> divisors(const T &n) {
    vector<T> ret;
    for (T i = 1; i * i <= n; i++) {
        if (n % i == 0) {
            ret.push_back(i);
            if (i * i != n) ret.push_back(n / i);
        }
    }
    sort(begin(ret), end(ret));
    return ret;
}
template <typename T>
vector<pair<T, int>> prime_factor(T n) {
    vector<pair<T, int>> ret;
    for (T i = 2; i * i <= n; i++) {
        int cnt = 0;
        while (n % i == 0) cnt++, n /= i;
        if (cnt > 0) ret.emplace_back(i, cnt);
    }
    if (n > 1) ret.emplace_back(n, 1);
    return ret;
}
template <typename T>
bool is_prime(const T &n) {
    if (n == 1) return false;
    for (T i = 2; i * i <= n; i++) {
        if (n % i == 0) return false;
    }
    return true;
}
// 1,2,...,n のうち k と互いに素である自然数の個数
template <typename T>
T coprime(T n, T k) {
    vector<pair<T, int>> ps = prime_factor(k);
    int m = ps.size();
    T ret = 0;
    for (int i = 0; i < (1 << m); i++) {
        T prd = 1;
        for (int j = 0; j < m; j++) {
            if ((i >> j) & 1) prd *= ps[j].first;
        }
        ret += (__builtin_parity(i) ? -1 : 1) * (n / prd);
    }
    return ret;
}
vector<bool> Eratosthenes(const int &n) {
    vector<bool> ret(n + 1, true);
    if (n >= 0) ret[0] = false;
    if (n >= 1) ret[1] = false;
    for (int i = 2; i * i <= n; i++) {
        if (!ret[i]) continue;
        for (int j = i + i; j <= n; j += i) ret[j] = false;
    }
    return ret;
}
vector<int> Eratosthenes2(const int &n) {
    vector<int> ret(n + 1);
    iota(begin(ret), end(ret), 0);
    if (n >= 0) ret[0] = -1;
    if (n >= 1) ret[1] = -1;
    for (int i = 2; i * i <= n; i++) {
        if (ret[i] < i) continue;
        for (int j = i + i; j <= n; j += i) ret[j] = min(ret[j], i);
    }
    return ret;
}
// n 以下の素数の数え上げ
template <typename T>
T count_prime(T n) {
    if (n < 2) return 0;
    vector<T> ns = {0};
    for (T i = n; i > 0; i = n / (n / i + 1)) ns.push_back(i);
    vector<T> h = ns;
    for (T &x : h) x--;
    for (T x = 2, m = sqrtl(n), k = ns.size(); x <= m; x++) {
        if (h[k - x] == h[k - x + 1]) continue; // h(x-1,x-1) = h(x-1,x) ならば x は素数ではない
        T x2 = x * x, pi = h[k - x + 1];
        for (T i = 1, j = ns[i]; i < k && j >= x2; j = ns[++i]) h[i] -= h[i * x <= m ? i * x : k - j / x] - pi;
    }
    return h[1];
}
// i 以下で i と互いに素な自然数の個数のテーブル
vector<int> Euler_totient_table(const int &n) {
    vector<int> dp(n + 1, 0);
    for (int i = 1; i <= n; i++) dp[i] = i;
    for (int i = 2; i <= n; i++) {
        if (dp[i] == i) {
            dp[i]--;
            for (int j = i + i; j <= n; j += i) {
                dp[j] /= i;
                dp[j] *= i - 1;
            }
        }
    }
    return dp;
}
// 約数包除に用いる係数テーブル (平方数で割り切れるなら 0、素因数の種類が偶数なら +1、奇数なら -1)
vector<int> inclusion_exclusion_table(int n) {
    auto p = Eratosthenes2(n);
    vector<int> ret(n + 1, 0);
    if (n >= 1) ret[1] = 1;
    for (int i = 2; i <= n; i++) {
        int x = p[i], j = i / x;
        ret[i] = (p[j] == x ? 0 : -ret[j]);
    }
    return ret;
}
using ld = long double;
void solve() {
    ll N, K;
    cin >> N >> K;
    auto sgn = inclusion_exclusion_table(N);
    auto calc = [&](ll x, ll y) {
        vector<ll> c(N + 1);
        rep2(i, 1, N + 1) {
            c[i] = (y == 0 ? INF : ld(x * i) / ld(y));
            chmin(c[i], N);
        }
        ll ret = 1;
        if (y == 0) ret++;
        rep2(i, 1, N + 1) {
            for (int j = i; j <= N; j += i) {
                ret += sgn[i] * floor(ld(c[j]) / ld(i)); //
            }
        }
        return ret - 1;
    };
    auto ch = [&](ld t) {
        ll mx = 0, my = 1;
        rep2(y, 1, N + 1) {
            ll x = floor(t * y);
            if (x * my >= y * mx) mx = x, my = y;
        }
        ll g = gcd(mx, my);
        mx /= g, my /= g;
        return pll(mx, my);
    };
    ll cnt = calc(1, 1);
    bool rev = false;
    if (K > 2 * cnt - 1) {
        cout << "-1\n";
        return;
    }
    if (K > cnt) {
        K = 2 * cnt - K;
        rev = true;
    }
    ld EPS = 1e-12;
    auto find = [&](ll K) {
        // int cnt = calc(1, 0);
        // if (cnt == K) return pll(1, 0);
        ld l = 0 - EPS, r = 1 + EPS;
        rep(_, 40) {
            ld m = (l + r) * 0.5;
            auto [x, y] = ch(m);
            (calc(x, y) >= K ? r : l) = m;
        }
        // cout << l << ' ' << r << '\n';
        r += EPS;
        return ch(r);
    };
    // cout << calc(1, N) << '\n';
    if (calc(N, 1) < K) {
        cout << "-1\n";
        return;
    }
    auto [x, y] = find(K);
    if (rev) swap(x, y);
    // assert(calc(x, y) == K);
    cout << x << "/" << y << '\n';
}
int main() {
    int T = 1;
    cin >> T;
    while (T--) solve();
}
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