AtCoder Beginner Contest 292

A - CAPS LOCK

#include <bits/stdc++.h>

using namespace std;

int32_t main() {
    string s;
    cin >> s;
    for( auto i : s )
        cout << char(i - 'a' + 'A');
    return 0;
}

B - Yellow and Red Card

把红牌当成两张黄牌，对于操作 3 就是查看\(x\)是否有大于等于两张黄牌

#include <bits/stdc++.h>

using namespace std;

int32_t main() {
    int n , q;
    cin >> n >> q;
    vector<int> a( n+1 , 0 );
    for( int opt , x ; q ; q -- ){
        cin >> opt >> x;
        if( opt == 1 ){
            a[x] ++;
        }else if( opt == 2 ){
            a[x] += 2;
        }else{
            cout << ( a[x] >= 2 ? "Yes\n" : "No\n");
        }
    }
    return 0;
}

C - Four Variables

令\(AB=x,CD=y\)则\(x+y=N,y=N-x\)，所以先\(O(n)\)枚举出\(x,y\)

然后在\(O(\sqrt{N})\)的枚举出\(A,B,C,D\)，记构成\(x\)的方案数是\(p\)，如果\(A=B\)，\(p=p+1\)，否则\(p=p+2\)，同理计算出\(y\)的方案数\(q\)

答案是\(res\)，如果\(x=y\)，\(res=res+pq\)，否则\(res=res+2pq\)

#include <bits/stdc++.h>

using namespace std;

#define int long long

int32_t main() {
    int n , res = 0;
    cin >> n;
    for( int a = 1 , b = n-1 , x , y ; a <= b ; a ++ , b -- ){
        x = y = 0;
        for( int i = 1 ; i * i <= a ; i ++ ){
            if( a % i ) continue;
            x ++;
            if( i != a / i ) x ++;
        }
        for( int i = 1 ; i * i <= b ; i ++ ){
            if( b % i ) continue;
            y ++;
            if( i != b / i ) y ++;
        }
        res += x * y;
        if( a != b ) res += x * y;
    }
    cout << res;
    return 0;
}

D - Unicyclic Components

并查集维护联通块，判断联通块内点数的二倍是否等于联通块内点度数之和

#include <bits/stdc++.h>

using namespace std;

int read(){
    int x = 0 , f = 1 , ch = getchar();
    while( (ch < '0' || ch > '9') && ch != '-' ) ch = getchar();
    if( ch == '-' ) f = -1 , ch = getchar();
    while( ch >= '0' && ch <= '9' ) x = ( x << 3 ) + ( x << 1 ) + ch - '0' , ch = getchar();
    return x * f;
}

class dsu{
private:
    vector<int> fa;
public:
    dsu( int n = 1 ){
        fa = vector<int>( n+1 , -1 ) , fa[0] = 0;
    }
    int getfa( int x ){
        if( fa[x] < 0 ) return x;
        return fa[x] = getfa( fa[x] );
    }
    void merge( int x , int y ){
        x = getfa(x) , y = getfa(y);
        if( x == y ) return ;
        if( fa[x] > fa[y] ) swap( x , y );
        fa[x] += fa[y] , fa[y] = x;
    }
    bool check( int x , int y ){
        x = getfa(x) , y = getfa(y);
        return ( x == y );
    }
    int size( int x ){
        x = getfa(x);
        return -fa[x];
    }
};

int32_t main() {
    int n = read() , m = read();
    vector<int> e(n+1);
    dsu d(n);
    for( int u , v ; m ; m -- ){
        u = read() , v = read();
        d.merge( u , v );
        e[u] ++ , e[v] ++;
    }
    map<int,int>cnt;
    for( int i = 1 , t; i <= n ; i ++ )
        t = d.getfa(i) , cnt[t] += e[i];
    for( auto [k,v] : cnt )
        if( v != d.size(k) * 2 ) return cout <<  "No\n" , 0;
    cout << "Yes\n";
    return 0;
}

E - Transitivity

对每一个点都进行 bfs，统计每一个点可以到达的所有的点。复杂度\(O(n^2)\)

#include<bits/stdc++.h>

using namespace std;


int32_t main() {
    int n , m , cnt;
    cin >> n >> m , cnt = -m;
    vector<vector<int>> e(n);
    for( int u, v; m; m -- )
        cin >> u >> v, u --, v --, e[u].emplace_back(v);
    for( int i = 0; i < n; i ++ ){
        vector<bool> vis(n,false);
        vis[i] = true;
        queue<int> q;
        q.push(i);
        while( !q.empty() ){
            auto u = q.front(); q.pop();
            for( auto v : e[u] )
                if( vis[v] == false )
                    vis[v] = true, q.push(v), cnt ++;
        }
    }
    cout << cnt;
}

F - Regular Triangle Inside a Rectangle

首先保证\(A<B\)，现在有两种情况

情况一是三角形的一条边和B重合此时答案就是\(r=\frac{A}{\sin\frac{\pi}{3}}\)，条件是\(A\le B\sin\frac{\pi}{3}\)

情况二就是三角形斜放，此时有\(r=\frac{A}{\cos\theta}=\frac{B}{\cos(\frac{\pi}{6}-\theta)}\)

\[A(\cos\frac{\pi}{6}\cos\theta=\sin\frac{\pi}{6}\sin\theta)=B\cos\theta\\ A(\frac{\sqrt{3}}{2}\cos\theta+\frac{1}{2}\sin\theta)=B\cos\theta\\ \frac{\sqrt 3 }{2} + \frac{\sin\theta}{2\cos\theta}=\frac B A\\ \tan\theta=\frac{2B}{A}-\sqrt 3 \]

所以\(\theta=\arctan(\frac{2B}{A}-\sqrt 3),r=\frac{A}{\cos\theta}\)

#include<bits/stdc++.h>

using namespace std;

typedef long double ld;

int32_t main() {
    ld a , b;
    const ld SQRT3 = sqrt(3);
    cin >> a >> b;
    if( a > b ) swap( a , b );         
    if( a <= b * SQRT3 / 2.0 )
        cout << fixed << setprecision(20) << a * 2.0 / SQRT3;
    else {
        ld theta = atan( 2*b/a - SQRT3 );
        cout << fixed << setprecision(20) << a / cos(theta);
    }
}