数学简单专题

http://vjudge.net/vjudge/contest/view.action?cid=52710#overview

简单推导

 1 #include <map>
 2 #include <set>
 3 #include <stack>
 4 #include <queue>
 5 #include <cmath>
 6 #include <ctime>
 7 #include <vector>
 8 #include <cstdio>
 9 #include <cctype>
10 #include <cstring>
11 #include <cstdlib>
12 #include <iostream>
13 #include <algorithm>
14 using namespace std;
15 #define INF 0x3f3f3f3f
16 #define MAX(a,b) (a > b ? a : b)
17 #define MIN(a,b) (a < b ? a : b)
18 #define mem0(a) memset(a,0,sizeof(a))
19 
20 //typedef long long LL;
21 typedef __int64 LL;
22 const double eps = 1e-12;
23 const int MAXN = 200005;
24 const int MAXM = 5005;
25 
26 int N;
27 
28 int calc()
29 {
30     if(N == 1) return 0;
31     if(N == 2) return 16;
32     int tot = (N*N*N) * (N*N*N-1) / 2;
33     int sub = 8 * 3 + 48 * (N-2) + 30 * (N-2) * (N-2) + (N-2)*(N-2)*(N-2) * 6;
34     return tot - sub / 2;
35 }
36 
37 int main()
38 {
39     //freopen("in.txt", "r", stdin);
40     //freopen("out.txt", "w", stdout);
41     while(~scanf("%d", &N))
42     {
43         printf("%d\n", calc());
44     }
45     return 0;
46 }

View Code

B - Points on Cycle HDU 1700

计算下，然后考虑特殊情况（精度开太高居然WA。。。。）

 1 #include <map>
 2 #include <set>
 3 #include <stack>
 4 #include <queue>
 5 #include <cmath>
 6 #include <ctime>
 7 #include <vector>
 8 #include <cstdio>
 9 #include <cctype>
10 #include <cstring>
11 #include <cstdlib>
12 #include <iostream>
13 #include <algorithm>
14 using namespace std;
15 #define INF 0x3f3f3f3f
16 #define inf (-((LL)1<<40))
17 #define lson k<<1, L, mid
18 #define rson k<<1|1, mid+1, R
19 #define mem0(a) memset(a,0,sizeof(a))
20 #define mem1(a) memset(a,-1,sizeof(a))
21 #define mem(a, b) memset(a, b, sizeof(a))
22 #define FOPENIN(IN) freopen(IN, "r", stdin)
23 #define FOPENOUT(OUT) freopen(OUT, "w", stdout)
24 template<class T> T CMP_MIN ( T a, T b ) { return a < b;   }
25 template<class T> T CMP_MAX ( T a, T b ) { return a > b;   }
26 template<class T> T MAX ( T a, T b ) { return a > b ? a : b; }
27 template<class T> T MIN ( T a, T b ) { return a < b ? a : b; }
28 template<class T> T GCD ( T a, T b ) { return b ? GCD ( b, a % b ) : a; }
29 template<class T> T LCM ( T a, T b ) { return a / GCD ( a, b ) * b;     }
30 template<class T> void SWAP( T& a, T& b ) { T t = a; a = b;  b = t;     }
31 
32 typedef __int64 LL;
33 //typedef long long LL;
34 const int MAXN = 100010;
35 const int MAXM = 110000;
36 const double eps = 1e-7;
37 const double PI = 4.0 * (double)atan(1.0);
38 
39 int main()
40 {
41     int T;
42     scanf("%d", &T);
43     while(T--)
44     {
45         double x, y;
46         scanf("%lf %lf", &x, &y);
47         double sita, r = sqrt(x * x + y * y);
48         if(fabs(x) < eps) {
49             if(y > 0) sita = PI / 2;
50             else sita = -PI / 2;
51         }
52         else sita = atan(y / x);
53         if(x < 0) sita += PI;
54 
55         double x1, y1, x2, y2;
56         sita += 2.0 * PI / 3.0;
57         x1 = r * cos(sita);
58         y1 = r * sin(sita);
59 
60         sita += 2.0 * PI / 3.0;
61         x2 = r * cos(sita);
62         y2 = r * sin(sita);
63 
64         if(y1 - y2 > eps) SWAP(x1, x2), SWAP(y1,y2);
65         else if(fabs(y1 - y2) < eps && x1 - x2 > eps) SWAP(x1, x2), SWAP(y1,y2);
66 
67         printf("%.3lf %.3lf %.3lf %.3lf\n", x1, y1, x2, y2);
68     }
69     return 0;
70 }

View Code

C - Eddy's 洗牌问题HDU 1210

模拟下洗牌过程

 1 #include <map>
 2 #include <set>
 3 #include <stack>
 4 #include <queue>
 5 #include <cmath>
 6 #include <ctime>
 7 #include <vector>
 8 #include <cstdio>
 9 #include <cctype>
10 #include <cstring>
11 #include <cstdlib>
12 #include <iostream>
13 #include <algorithm>
14 using namespace std;
15 #define INF 0x3f3f3f3f
16 #define inf (-((LL)1<<40))
17 #define lson k<<1, L, mid
18 #define rson k<<1|1, mid+1, R
19 #define mem0(a) memset(a,0,sizeof(a))
20 #define mem1(a) memset(a,-1,sizeof(a))
21 #define mem(a, b) memset(a, b, sizeof(a))
22 #define FOPENIN(IN) freopen(IN, "r", stdin)
23 #define FOPENOUT(OUT) freopen(OUT, "w", stdout)
24 template<class T> T CMP_MIN ( T a, T b ) { return a < b;   }
25 template<class T> T CMP_MAX ( T a, T b ) { return a > b;   }
26 template<class T> T MAX ( T a, T b ) { return a > b ? a : b; }
27 template<class T> T MIN ( T a, T b ) { return a < b ? a : b; }
28 template<class T> T GCD ( T a, T b ) { return b ? GCD ( b, a % b ) : a; }
29 template<class T> T LCM ( T a, T b ) { return a / GCD ( a, b ) * b;     }
30 template<class T> void SWAP( T& a, T& b ) { T t = a; a = b;  b = t;     }
31 
32 typedef __int64 LL;
33 //typedef long long LL;
34 const int MAXN = 100010;
35 const int MAXM = 110000;
36 const double eps = 1e-14;
37 const double PI = 4.0 * (double)atan(1.0);
38 
39 int main()
40 {
41     int N;
42     while(~scanf("%d", &N))
43     {
44         int ans = 0, x = 1;
45         do
46         {
47             if(x <= N) x <<= 1;
48             else x = (x-N)*2-1;
49             ans ++;
50         }while(x != 1);
51         printf("%d\n", ans);
52     }
53     return 0;
54 }

View Code

D - Delta-wave HDU 1030

算出水平方向，左斜方向，右斜方向的所处的层数，相减取绝对值即可

 1 #include <map>
 2 #include <set>
 3 #include <stack>
 4 #include <queue>
 5 #include <cmath>
 6 #include <ctime>
 7 #include <vector>
 8 #include <cstdio>
 9 #include <cctype>
10 #include <cstring>
11 #include <cstdlib>
12 #include <iostream>
13 #include <algorithm>
14 using namespace std;
15 #define INF 0x3f3f3f3f
16 #define inf (-((LL)1<<40))
17 #define lson k<<1, L, mid
18 #define rson k<<1|1, mid+1, R
19 #define mem0(a) memset(a,0,sizeof(a))
20 #define mem1(a) memset(a,-1,sizeof(a))
21 #define mem(a, b) memset(a, b, sizeof(a))
22 #define FOPENIN(IN) freopen(IN, "r", stdin)
23 #define FOPENOUT(OUT) freopen(OUT, "w", stdout)
24 template<class T> T ABS ( T a) { return a >= 0 ? a : -a;   }
25 template<class T> T CMP_MIN ( T a, T b ) { return a < b;   }
26 template<class T> T CMP_MAX ( T a, T b ) { return a > b;   }
27 template<class T> T MAX ( T a, T b ) { return a > b ? a : b; }
28 template<class T> T MIN ( T a, T b ) { return a < b ? a : b; }
29 template<class T> T GCD ( T a, T b ) { return b ? GCD ( b, a % b ) : a; }
30 template<class T> T LCM ( T a, T b ) { return a / GCD ( a, b ) * b;     }
31 template<class T> void SWAP( T& a, T& b ) { T t = a; a = b;  b = t;     }
32 
33 typedef __int64 LL;
34 //typedef long long LL;
35 const int MAXN = 100010;
36 const int MAXM = 110000;
37 const double eps = 1e-14;
38 const double PI = 4.0 * (double)atan(1.0);
39 
40 void get(LL &x, LL &y, LL &z, int n)
41 {
42     for(x=1;x*x<n;x++);
43     y = (n - (x-1)*(x-1) - 1) / 2 + 1;
44     z = (x*x - n) / 2 + 1;
45 }
46 
47 int main()
48 {
49     int N, M;
50     while(~scanf("%d %d", &N, &M))
51     {
52         LL ax, ay, az, bx, by, bz;
53         get(ax, ay, az, N);
54         get(bx, by, bz, M);
55         printf("%I64d\n", ABS(ax-bx) + ABS(ay-by) + ABS(az-bz));
56     }
57     return 0;
58 }

View Code

F - 猜数字 HDU 2178

 1 #include <map>
 2 #include <set>
 3 #include <stack>
 4 #include <queue>
 5 #include <cmath>
 6 #include <ctime>
 7 #include <vector>
 8 #include <cstdio>
 9 #include <cctype>
10 #include <cstring>
11 #include <cstdlib>
12 #include <iostream>
13 #include <algorithm>
14 using namespace std;
15 #define INF 0x3f3f3f3f
16 #define MAX(a,b) (a > b ? a : b)
17 #define MIN(a,b) (a < b ? a : b)
18 #define mem0(a) memset(a,0,sizeof(a))
19 
20 //typedef long long LL;
21 typedef __int64 LL;
22 const double eps = 1e-12;
23 const int MAXN = 200005;
24 const int MAXM = 5005;
25 
26 
27 int main()
28 {
29     //freopen("in.txt", "r", stdin);
30     //freopen("out.txt", "w", stdout);
31     int t, N;
32     while(~scanf("%d", &t))while(t--)
33     {
34         scanf("%d", &N);
35         printf("%d\n", (1<<N)-1);
36     }
37     return 0;
38 }

View Code

G - Catching the ThiefHDU 3469

由于警察是足够聪明的，所以他可以选择从第二个城市开始一直到倒数第二个结束每个呆两天（可以保证左边绝对不会有小偷）

 1 #include <map>
 2 #include <set>
 3 #include <stack>
 4 #include <queue>
 5 #include <cmath>
 6 #include <ctime>
 7 #include <vector>
 8 #include <cstdio>
 9 #include <cctype>
10 #include <cstring>
11 #include <cstdlib>
12 #include <iostream>
13 #include <algorithm>
14 using namespace std;
15 #define INF 0x3f3f3f3f
16 #define inf (-((LL)1<<40))
17 #define lson k<<1, L, mid
18 #define rson k<<1|1, mid+1, R
19 #define mem0(a) memset(a,0,sizeof(a))
20 #define mem1(a) memset(a,-1,sizeof(a))
21 #define mem(a, b) memset(a, b, sizeof(a))
22 #define FOPENIN(IN) freopen(IN, "r", stdin)
23 #define FOPENOUT(OUT) freopen(OUT, "w", stdout)
24 template<class T> T ABS ( T a) { return a >= 0 ? a : -a;   }
25 template<class T> T CMP_MIN ( T a, T b ) { return a < b;   }
26 template<class T> T CMP_MAX ( T a, T b ) { return a > b;   }
27 template<class T> T MAX ( T a, T b ) { return a > b ? a : b; }
28 template<class T> T MIN ( T a, T b ) { return a < b ? a : b; }
29 template<class T> T GCD ( T a, T b ) { return b ? GCD ( b, a % b ) : a; }
30 template<class T> T LCM ( T a, T b ) { return a / GCD ( a, b ) * b;     }
31 template<class T> void SWAP( T& a, T& b ) { T t = a; a = b;  b = t;     }
32 
33 typedef __int64 LL;
34 //typedef long long LL;
35 const int MAXN = 100010;
36 const int MAXM = 110000;
37 const double eps = 1e-14;
38 const double PI = 4.0 * (double)atan(1.0);
39 
40 int main()
41 {
42     int N, t, T=0;
43     while(~scanf("%d", &t))while(t--)
44     {
45         scanf("%d", &N);
46         int ans;
47         if(N == 1) ans = 1;
48         else if(N == 2) ans = 2;
49         else ans = (N - 2) * 2;
50         printf("Case %d: %d\n", ++T, ans);
51     }
52     return 0;
53 }

View Code

H - Combinations, Once AgainPOJ 1285

建立状态F[i][j]表示第i个数选择j个的总方案数，可以枚举第i个数选了多少个，那么

F[i][j] = sum{DP[i-1][j-k] | k<=min(n[i], j) }

 1 #include <map>
 2 #include <set>
 3 #include <stack>
 4 #include <queue>
 5 #include <cmath>
 6 #include <ctime>
 7 #include <vector>
 8 #include <cstdio>
 9 #include <cctype>
10 #include <cstring>
11 #include <cstdlib>
12 #include <iostream>
13 //#include <algorithm>
14 using namespace std;
15 #define INF 0x3f3f3f3f
16 #define inf (-((LL)1<<40))
17 #define lson k<<1, L, mid
18 #define rson k<<1|1, mid+1, R
19 #define mem0(a) memset(a,0,sizeof(a))
20 #define mem1(a) memset(a,-1,sizeof(a))
21 #define mem(a, b) memset(a, b, sizeof(a))
22 #define FOPENIN(IN) freopen(IN, "r", stdin)
23 #define FOPENOUT(OUT) freopen(OUT, "w", stdout)
24 template<class T> T ABS ( T a) { return a >= 0 ? a : -a;   }
25 template<class T> T CMP_MIN ( T a, T b ) { return a < b;   }
26 template<class T> T CMP_MAX ( T a, T b ) { return a > b;   }
27 template<class T> T MAX ( T a, T b ) { return a > b ? a : b; }
28 template<class T> T MIN ( T a, T b ) { return a < b ? a : b; }
29 template<class T> T GCD ( T a, T b ) { return b ? GCD ( b, a % b ) : a; }
30 template<class T> T LCM ( T a, T b ) { return a / GCD ( a, b ) * b;     }
31 template<class T> void SWAP( T& a, T& b ) { T t = a; a = b;  b = t;     }
32 
33 typedef __int64 LL;
34 //typedef long long LL;
35 const int MAXN = 100;
36 const int MAXM = 110000;
37 const double eps = 1e-14;
38 const double PI = 4.0 * atan(1.0);
39 
40 LL DP[MAXN][MAXN];
41 int n[MAXN];
42 int N, M;
43 
44 int main()
45 {
46     int cas = 0;
47     while(~scanf("%d%d", &N, &M) && (N || M))
48     {
49         int x;
50         mem0(n);
51         for(int i = 0; i < N; i ++ )
52         {
53             scanf("%d", &x);
54             n[x] ++;
55         }
56         mem0(DP);
57         for(int i = 0; i <= N; i ++)
58             DP[i][0] = 1;
59         for(int i = 1; i<= N; i ++)
60         {
61             for(int j = 1; j <= N; j ++)
62             {
63                 for(int k = 0; k <= n[i] && k <= j; k ++)
64                     DP[i][j] += DP[i-1][j-k];
65             }
66         }
67         printf("Case %d:\n", ++cas);
68         while(M--)
69         {
70             scanf("%d", &x);
71             printf("%I64d\n", DP[N][x]);
72         }
73     }
74     return 0;
75 }

View Code

I - Self NumbersPOJ 1316

模拟一遍即可

 1 #include <map>
 2 #include <set>
 3 #include <stack>
 4 #include <queue>
 5 #include <cmath>
 6 #include <ctime>
 7 #include <vector>
 8 #include <cstdio>
 9 #include <cctype>
10 #include <cstring>
11 #include <cstdlib>
12 #include <iostream>
13 //#include <algorithm>
14 using namespace std;
15 #define INF 0x3f3f3f3f
16 #define inf (-((LL)1<<40))
17 #define lson k<<1, L, mid
18 #define rson k<<1|1, mid+1, R
19 #define mem0(a) memset(a,0,sizeof(a))
20 #define mem1(a) memset(a,-1,sizeof(a))
21 #define mem(a, b) memset(a, b, sizeof(a))
22 #define FOPENIN(IN) freopen(IN, "r", stdin)
23 #define FOPENOUT(OUT) freopen(OUT, "w", stdout)
24 template<class T> T ABS ( T a) { return a >= 0 ? a : -a;   }
25 template<class T> T CMP_MIN ( T a, T b ) { return a < b;   }
26 template<class T> T CMP_MAX ( T a, T b ) { return a > b;   }
27 template<class T> T MAX ( T a, T b ) { return a > b ? a : b; }
28 template<class T> T MIN ( T a, T b ) { return a < b ? a : b; }
29 template<class T> T GCD ( T a, T b ) { return b ? GCD ( b, a % b ) : a; }
30 template<class T> T LCM ( T a, T b ) { return a / GCD ( a, b ) * b;     }
31 template<class T> void SWAP( T& a, T& b ) { T t = a; a = b;  b = t;     }
32 
33 typedef __int64 LL;
34 //typedef long long LL;
35 const int MAXN = 10005;
36 const int MAXM = 110000;
37 const double eps = 1e-14;
38 const double PI = 4.0 * atan(1.0);
39 
40 int vis[MAXN];
41 
42 int sum(int x)
43 {
44     int ans = 0;
45     while(x) ans += x%10, x /= 10;
46     return ans;
47 }
48 
49 int main()
50 {
51     //FOPENIN("in.txt");
52     for(int i = 1; i <= 10000; i ++)
53     {
54         int x = i;
55         do
56         {
57             x += sum(x);
58             vis[x] = 1;
59         }while(x <= 10000 && !vis[x]);
60     }
61     for(int i=1;i<=10000;i++)if(!vis[i])
62     {
63         printf("%d\n", i);
64     }
65     return 0;
66 }

View Code

J - Prime Test POJ 1811

米勒拉宾素数测试

  1 #include <map>
  2 #include <set>
  3 #include <stack>
  4 #include <queue>
  5 #include <cmath>
  6 #include <ctime>
  7 #include <vector>
  8 #include <cstdio>
  9 #include <cctype>
 10 #include <cstring>
 11 #include <cstdlib>
 12 #include <iostream>
 13 //#include <algorithm>
 14 using namespace std;
 15 #define INF 0x3f3f3f3f
 16 #define inf ((LL)1<<40)
 17 #define lson k<<1, L, mid
 18 #define rson k<<1|1, mid+1, R
 19 #define mem0(a) memset(a,0,sizeof(a))
 20 #define mem1(a) memset(a,-1,sizeof(a))
 21 #define mem(a, b) memset(a, b, sizeof(a))
 22 #define FOPENIN(IN) freopen(IN, "r", stdin)
 23 #define FOPENOUT(OUT) freopen(OUT, "w", stdout)
 24 template<class T> T ABS ( T a) { return a >= 0 ? a : -a;   }
 25 template<class T> T CMP_MIN ( T a, T b ) { return a < b;   }
 26 template<class T> T CMP_MAX ( T a, T b ) { return a > b;   }
 27 template<class T> T MAX ( T a, T b ) { return a > b ? a : b; }
 28 template<class T> T MIN ( T a, T b ) { return a < b ? a : b; }
 29 template<class T> T GCD ( T a, T b ) { return b ? GCD ( b, a % b ) : a; }
 30 template<class T> T LCM ( T a, T b ) { return a / GCD ( a, b ) * b;     }
 31 template<class T> void SWAP( T& a, T& b ) { T t = a; a = b;  b = t;     }
 32 
 33 typedef __int64 LL;
 34 //typedef long long LL;
 35 const int MAXN = 10005;
 36 const int MAXM = 110000;
 37 const double eps = 1e-14;
 38 const double PI = 4.0 * atan(1.0);
 39 
 40 
 41 
 42 
 43 LL n, ans;
 44 int t;
 45 
 46 
 47 
 48 
 49 #define Times 10
 50 
 51 //产生[0, n-1]的一个随机数
 52 LL random(LL n)
 53 {
 54     return ((double)rand() / RAND_MAX * n + 0.5);
 55 }
 56 //乘法，采用加法模拟，避免中间结果超出LL
 57 LL multi(LL a,LL b,LL mod)
 58 {
 59     LL ans=0;
 60     while(b)
 61     {
 62         if(b & 1)
 63         {
 64             b--;
 65             ans=(ans+a) % mod;
 66         }
 67         else
 68         {
 69             b/=2;
 70             a=(2*a) % mod;
 71         }
 72     }
 73     return ans;
 74 }
 75 
 76 //快速幂，同样避免超出LL的做法
 77 LL Pow(LL a, LL b, LL mod)
 78 {
 79     LL ans=1;
 80     while(b)
 81     {
 82         if(b&1)
 83         {
 84             b--;
 85             ans=multi(ans,a,mod);
 86         }
 87         else
 88         {
 89             b/=2;
 90             a=multi(a,a,mod);
 91         }
 92     }
 93     return ans;
 94 }
 95 
 96 //miller_rabin的一遍探测，返回false表示是合数
 97 bool witness(LL a,LL n)
 98 {
 99     LL d=n-1;
100     while( !(d&1) )
101         d >>= 1;
102     LL t=Pow(a, d, n);
103     while(d!=n-1 && t!=1 && t!=n-1)
104     {
105         t=multi(t,t,n);
106         d<<=1;
107     }
108     return t==n-1 || d&1;
109 }
110 
111 //miller_rabin算法，返回false表示是合数，否则是素数
112 //返回素数出错的概率(最高)为 1 / (4 ^ times)
113 bool miller_rabin(LL n)
114 {
115     if(n == 2)
116         return true;
117     if( n<2 || !(n&1) )
118         return false;
119     for(int i = 1; i <= Times ; i++ )
120     {
121         LL a = random(n-2) + 1;
122         if( !witness(a, n) )
123             return false;
124     }
125     return true;
126 }
127 
128 
129 //************************************************
130 //pollard_rho 算法进行质因数分解
131 //************************************************
132 LL factor[100];//质因数分解结果（刚返回时是无序的）
133 int tol;//质因数的个数。数组小标从0开始
134 
135 LL gcd(LL a,LL b)
136 {
137     if(a==0)return 1;//???????
138     if(a<0) return gcd(-a,b);
139     while(b)
140     {
141         LL t=a%b;
142         a=b;
143         b=t;
144     }
145     return a;
146 }
147 
148 LL Pollard_rho(LL x, LL c)
149 {
150     LL i=1,k=2;
151     LL x0=rand()%x;
152     LL y=x0;
153     while(1)
154     {
155         i++;
156         x0=(multi(x0, x0, x) + c) % x;
157         LL d=gcd(y-x0,x);
158         if(d!=1&&d!=x) return d;
159         if(y==x0) return x;
160         if(i==k){y=x0;k+=k;}
161     }
162 }
163 
164 //对n进行素因子分解
165 void findfac(LL n)
166 {
167     if(miller_rabin(n))//素数
168     {
169         factor[tol++]=n;
170         ans = MIN(ans, n);
171         return;
172     }
173     LL p=n;
174     while(p>=n)
175         p=Pollard_rho(p, rand()%(n-1)+1);
176     findfac(p);
177     findfac(n/p);
178 }
179 
180 
181 
182 int main()
183 {
184     //FOPENIN("in.txt");
185     while(~scanf("%d", &t))while(t--)
186     {
187         ans = inf;
188         scanf("%I64d", &n);
189         findfac(n);
190         if(miller_rabin(n))
191             printf("Prime\n");
192         else printf("%I64d\n", ans);
193     }
194     return 0;
195 }

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posted @ 2014-08-21 00:11 再见~雨泉阅读(219) 评论(0) 编辑收藏举报

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