Codeforces Global Round 8
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：
- \(\frac{x+k}{k-1}+x+k\leq n\)
求解得\(x\leq n-\frac{n}{k}-k\)。所以\(x\)至多为\(n-\lceil\frac{n}{k}\rceil-k+1\)。
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 << '\n';
}
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 << '\n';
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) << '\n';
for (auto it : ans) {
cout << it.fi << ' ' << it.se << '\n';
}
}
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 << '\n';
}
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) << '\n';
for (auto it : ans) {
cout << it << ' ';
}
cout << '\n';
}
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;
}
重要的是自信，一旦有了自信，人就会赢得一切。
