Codeforces Global Round 8

传送门

视频题解

A. C+=

显然选择增加较小的一个数更优。

B. Codeforces Subsequences

贪心构造即可，显然个数均分能使得最后乘积最大，因为要长度最小，所以枚举最小长度贪心就行。

C. Even Picture

类似于这样构造：

D. AND, OR and square sum

操作的本质相当于交换(0,1)，显然所有的(1)尽可能跑到较大的数上面即可。

E. Ski Accidents

要用好每个点的出边为(2)并且删除总点数不超过(frac{4}{7})这个条件。
考虑一颗三层的完全二叉树，总结点数为(7)，第三层的结点数为(4)，刚好符合条件。
所以做法就是删除所有“第三层”的结点即可，容易证明“第三层”的结点不超过总结点的(frac{4}{7})。
具体做法就是按照拓扑序来求到达每个点的最长路来判断是否在“第三层”。

F. Lamps on a Circle

首先考虑如何求(R(n))，假设我们现在亮着(x)个灯泡，并且想要通过再点亮(k)个灯泡使得(x)增加，那么此时连续的亮着的段长度不超过(k-1)，即至少(frac{x+k}{k-1})段。
那么暗着的段也至少(frac{x+k}{k-1})段，故有不等式：

• (frac{x+k}{k-1}+x+kleq n)

求解得(xleq n-frac{n}{k}-k)。所以(x)至多为(n-lceilfrac{n}{k} ceil-k+1)。
找到(R(n))过后接下来就是构造最优方案，发现我们每次都使得连续亮着灯泡长度不超过(k-1)就行，所以将环(k)个(k)个进行分割，并且每段第(k)个标记不可选，这样就能满足条件，每次操作过后亮着的灯泡都会增加，直到(geq R(n))就行。

---

A. C+=

/*
 * Author:  heyuhhh
 * Created Time:  2020/6/18 22:46:31
 */
#include <iostream>
#include <algorithm>
#include <cstring>
#include <cstdio>
#include <vector>
#include <cmath>
#include <set>
#include <map>
#include <queue>
#include <iomanip>
#include <assert.h>
#include <functional>
#include <numeric>
#define MP make_pair
#define fi first
#define se second
#define pb push_back
#define sz(x) (int)(x).size()
#define all(x) (x).begin(), (x).end()
#define INF 0x3f3f3f3f
#define Local
#ifdef Local
  #define dbg(args...) do { cout << #args << " -> "; err(args); } while (0)
  void err() { std::cout << std::endl; }
  template<typename T, typename...Args>
  void err(T a, Args...args) { std::cout << a << ' '; err(args...); }
  template <template<typename...> class T, typename t, typename... A> 
  void err(const T <t> &arg, const A&... args) {
  for (auto &v : arg) std::cout << v << ' '; err(args...); }
#else
  #define dbg(...)
#endif
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
//head
const int N = 1e5 + 5;
 
void run() {
    int a, b, n;
    cin >> a >> b >> n;
    int ans = 0;
    while (a <= n && b <= n) {
        if (a < b) a += b;
        else b += a;
        ++ans;
    }
    cout << ans << '
';
}
 
int main() {
    ios::sync_with_stdio(false);
    cin.tie(0); cout.tie(0);
    cout << fixed << setprecision(20);
    int T; cin >> T; while(T--)
    run();
    return 0;
}

B. Codeforces Subsequences

/*
 * Author:  heyuhhh
 * Created Time:  2020/6/18 22:51:51
 */
#include <iostream>
#include <algorithm>
#include <cstring>
#include <cstdio>
#include <vector>
#include <cmath>
#include <set>
#include <map>
#include <queue>
#include <iomanip>
#include <assert.h>
#include <functional>
#include <numeric>
#define MP make_pair
#define fi first
#define se second
#define pb push_back
#define sz(x) (int)(x).size()
#define all(x) (x).begin(), (x).end()
#define INF 0x3f3f3f3f
#define Local
#ifdef Local
  #define dbg(args...) do { cout << #args << " -> "; err(args); } while (0)
  void err() { std::cout << std::endl; }
  template<typename T, typename...Args>
  void err(T a, Args...args) { std::cout << a << ' '; err(args...); }
  template <template<typename...> class T, typename t, typename... A> 
  void err(const T <t> &arg, const A&... args) {
  for (auto &v : arg) std::cout << v << ' '; err(args...); }
#else
  #define dbg(...)
#endif
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
//head
const int N = 1e5 + 5;
 
ll qpow(ll a, ll b) {
    ll res = 1;
    while (b) {
        if (b & 1) res = res * a;
        a = a * a;
        b >>= 1;
    }
    return res;   
}
 
const string str = "codeforces";
 
void run() {
    long long k; cin >> k;
    int t;
    for (int i = 1;;i++) {
        if (qpow(i, 10) > k) {
            t = i - 1;
            break;
        }
    }
    for (int i = 0; i <= 10; i++) {
        ll res = 1;
        for (int j = 1; j <= i; j++) {
            res *= (t + 1);
        }
        for (int j = 1; j <= 10 - i; j++) {
            res *= (t);
        }
        if (res >= k) {
            string res = "";
            int now = 0;
            for (int j = 1; j <= i; j++) {
                res += string(t + 1, str[now++]);
            }
            for (int j = 1; j <= 10 - i; j++) {
                res += string(t, str[now++]);
            }
            cout << res << '
';
            return;
        }
    }
}
 
int main() {
    ios::sync_with_stdio(false);
    cin.tie(0); cout.tie(0);
    cout << fixed << setprecision(20);
    run();
    return 0;
}

C. Even Picture

/*
 * Author:  heyuhhh
 * Created Time:  2020/6/18 23:14:47
 */
#include <iostream>
#include <algorithm>
#include <cstring>
#include <cstdio>
#include <vector>
#include <cmath>
#include <set>
#include <map>
#include <queue>
#include <iomanip>
#include <assert.h>
#include <functional>
#include <numeric>
#define MP make_pair
#define fi first
#define se second
#define pb push_back
#define sz(x) (int)(x).size()
#define all(x) (x).begin(), (x).end()
#define INF 0x3f3f3f3f
#define Local
#ifdef Local
  #define dbg(args...) do { cout << #args << " -> "; err(args); } while (0)
  void err() { std::cout << std::endl; }
  template<typename T, typename...Args>
  void err(T a, Args...args) { std::cout << a << ' '; err(args...); }
  template <template<typename...> class T, typename t, typename... A> 
  void err(const T <t> &arg, const A&... args) {
  for (auto &v : arg) std::cout << v << ' '; err(args...); }
#else
  #define dbg(...)
#endif
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
//head
const int N = 1e5 + 5;
 
void run() {
    int n; cin >> n;
    int x = n;
    int nx = 0, ny = 0;
    vector <pii> ans;
    auto draw = [&] (int x, int y, int op) {
        if (op) ans.push_back(MP(x, y));
        ans.push_back(MP(x, y - 1));
        if (op) ans.push_back(MP(x, y + 1));
        if (op) ans.push_back(MP(x - 1, y));
        if (op) ans.push_back(MP(x - 1, y + 1));
        ans.push_back(MP(x + 1, y));
        ans.push_back(MP(x + 1, y - 1));
        return 1;
    };
    draw(nx, ny, 1), --n, ++nx, --ny;
    for (int i = n; i >= 1; i--) {
        draw(nx, ny, 0);
        ++nx, --ny;
    }
    cout << sz(ans) << '
';
    for (auto it : ans) {
        cout << it.fi << ' ' << it.se << '
';
    }
}
 
int main() {
    ios::sync_with_stdio(false);
    cin.tie(0); cout.tie(0);
    cout << fixed << setprecision(20);
    run();
    return 0;
}

D. AND, OR and square sum

/*
 * Author:  heyuhhh
 * Created Time:  2020/6/18 23:56:37
 */
#include <iostream>
#include <algorithm>
#include <cstring>
#include <cstdio>
#include <vector>
#include <cmath>
#include <set>
#include <map>
#include <queue>
#include <iomanip>
#include <assert.h>
#include <functional>
#include <numeric>
#define MP make_pair
#define fi first
#define se second
#define pb push_back
#define sz(x) (int)(x).size()
#define all(x) (x).begin(), (x).end()
#define INF 0x3f3f3f3f
#define Local
#ifdef Local
  #define dbg(args...) do { cout << #args << " -> "; err(args); } while (0)
  void err() { std::cout << std::endl; }
  template<typename T, typename...Args>
  void err(T a, Args...args) { std::cout << a << ' '; err(args...); }
  template <template<typename...> class T, typename t, typename... A> 
  void err(const T <t> &arg, const A&... args) {
  for (auto &v : arg) std::cout << v << ' '; err(args...); }
#else
  #define dbg(...)
#endif
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
//head
const int N = 1e5 + 5;
 
void run() {
    int n; cin >> n;
    vector <int> a(n);
    vector <int> cnt(20);
    for (int i = 0; i < n; i++) {
        cin >> a[i];
        for (int j = 0; j < 20; j++) {
            if (a[i] >> j & 1) {
                ++cnt[j];
            }
        }
    }
    vector <int> b(n);
    for (int i = 0; i < 20; i++) {
        for (int j = 0; j < cnt[i]; j++) {
            b[j] += (1 << i);
        }
    }
    ll ans = 0;
    for (int i = 0; i < n; i++) {
        ans += 1ll * b[i] * b[i];
    }
    cout << ans << '
';
}
 
int main() {
    ios::sync_with_stdio(false);
    cin.tie(0); cout.tie(0);
    cout << fixed << setprecision(20);
    run();
    return 0;
}

E. Ski Accidents

/*
 * Author:  heyuhhh
 * Created Time:  2020/6/19 8:34:31
 */
#include <iostream>
#include <algorithm>
#include <cstring>
#include <cstdio>
#include <vector>
#include <cmath>
#include <set>
#include <map>
#include <queue>
#include <iomanip>
#include <assert.h>
#include <functional>
#include <numeric>
#define MP make_pair
#define fi first
#define se second
#define pb push_back
#define sz(x) (int)(x).size()
#define all(x) (x).begin(), (x).end()
#define INF 0x3f3f3f3f
#define Local
#ifdef Local
  #define dbg(args...) do { cout << #args << " -> "; err(args); } while (0)
  void err() { std::cout << std::endl; }
  template<typename T, typename...Args>
  void err(T a, Args...args) { std::cout << a << ' '; err(args...); }
  template <template<typename...> class T, typename t, typename... A> 
  void err(const T <t> &arg, const A&... args) {
  for (auto &v : arg) std::cout << v << ' '; err(args...); }
#else
  #define dbg(...)
#endif
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
//head
const int N = 2e5 + 5;
 
vector <int> rG[N];
int n, m;
 
void run() {
    cin >> n >> m;
    for (int i = 1; i <= n; i++) {
        rG[i].clear();
    }
    for (int i = 1; i <= m; i++) {
        int u, v; cin >> u >> v;
        rG[v].push_back(u);
    }
    vector <int> path(n + 1, 0);
    vector <int> ans;
    for (int i = 1; i <= n; i++) {
        for (auto j : rG[i]) {
            path[i] = max(path[j] + 1, path[i]);
        }
        if (path[i] == 2) {
            path[i] = -1;
            ans.push_back(i);
        }
    }
    cout << sz(ans) << '
';
    for (auto it : ans) {
        cout << it << ' ';
    }
    cout << '
';
}
 
int main() {
    ios::sync_with_stdio(false);
    cin.tie(0); cout.tie(0);
    cout << fixed << setprecision(20);
    int T; cin >> T; while(T--)
    run();
    return 0;
}

F. Lamps on a Circle

/*
 * Author:  heyuhhh
 * Created Time:  2020/6/19 1:01:08
 */
#include <iostream>
#include <algorithm>
#include <cstring>
#include <cstdio>
#include <vector>
#include <cmath>
#include <set>
#include <map>
#include <queue>
#include <iomanip>
#include <assert.h>
#include <functional>
#include <numeric>
#define MP make_pair
#define fi first
#define se second
#define pb push_back
#define sz(x) (int)(x).size()
#define all(x) (x).begin(), (x).end()
#define INF 0x3f3f3f3f
#define Local
#ifdef Local
  #define dbg(args...) do { cout << #args << " -> "; err(args); } while (0)
  void err() { std::cout << std::endl; }
  template<typename T, typename...Args>
  void err(T a, Args...args) { std::cout << a << ' '; err(args...); }
  template <template<typename...> class T, typename t, typename... A> 
  void err(const T <t> &arg, const A&... args) {
  for (auto &v : arg) std::cout << v << ' '; err(args...); }
#else
  #define dbg(...)
#endif
using namespace std;
typedef long long ll;
typedef pair<int, int> pii;
//head
const int N = 1e3 + 5;
 
int n;
 
int query(vector <int>& a) {
    cout << sz(a);
    for (auto it : a) {
        cout << ' ' << it;
    }
    cout << endl;
    int s; cin >> s;
    return s;
}
 
int answer() {
    cout << 0 << endl;
    exit(0);;
}
 
void run() {
    cin >> n;
    if (n <= 3) {
        answer();
    }
    int k, Max = -1;
    for (int i = 1; i <= n; i++) {
        if (n - i - (n + i - 1) / i + 1 > Max) {
            Max = n - i - (n + i - 1) / i + 1;
            k = i;
        }
    }
    vector <bool> ban(n + 1);
    for (int i = k; i <= n; i += k) {
        ban[i] = true;
    }
    vector <int> on(n + 1);
    while (1) {
        if (count(all(on), 1) >= Max) {
            answer();
        }
        int cnt = k;
        vector <int> q;
        for (int i = 1; i <= n && cnt; i++) {
            if (!ban[i] && !on[i]) {
                on[i] = 1;
                q.push_back(i);
                --cnt;
            }
        }
        int s = query(q);
        cnt = k;
        while (cnt--) {
            on[s] = 0;
            ++s;
            if (s > n) s -= n;
        }
    }
}
 
int main() {
    ios::sync_with_stdio(false);
    cin.tie(0); cout.tie(0);
    cout << fixed << setprecision(20);
    run();
    return 0;
}