数学基础

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title: 数学基础
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基本计数方法

Chess QueenUVA - 11538 (平方和公式由(n+1)^3推出)

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod=998244353;
const int maxn=1e6+10;
const ll inf=0x3f3f3f3f3f3f3f3fLL;

int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    std::cout.tie(0);
    ll n,m;
    while(cin>>n>>m&&n&&m){
        if(n>m)swap(n,m);
        cout<<n*m*(m-1)+n*m*(n-1)+2*n*(n-1)*(3*m-n-1)/3<<endl;
    }
    return 0;
}

Triangle CountingUVA - 11401 (从1-n中取出三个不同整数组成三角形的方法)

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod=998244353;
const int maxn=1e6+10;
const ll inf=0x3f3f3f3f3f3f3f3fLL;
ll dp[maxn];
int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    std::cout.tie(0);
    dp[3]=0;
    for(ll x=4;x<=1e6;x++)
        dp[x]=dp[x-1]+((x-1)*(x-2)/2-(x-1)/2)/2;
    int n;
    while(cin>>n){
        if(n<3)break;
        cout<<dp[n]<<endl;
    }
    return 0;
}

Triangle CountingUVA - 11401（容斥定理）

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod=1e6+7;
const int maxn=1e6+10;
const ll inf=0x3f3f3f3f3f3f3f3fLL;
int c[550][550];

int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    std::cout.tie(0);
    c[0][0]=1;
    for(int i=0;i<=500;i++){
        c[i][0]=c[i][i]=1;
        for(int j=1;j<i;j++)
            c[i][j]=(c[i-1][j]+c[i-1][j-1])%mod;
    }
    int t;
    cin>>t;
    for(int cas=1;cas<=t;cas++){
        int n,m,k,sum=0;
        cin>>n>>m>>k;
        for(int sta=0;sta<(1<<4);sta++){
            int b=0,r=n,cc=m;
            if(sta&1)r--,b++;
            if(sta&2)r--,b++;
            if(sta&4)cc--,b++;
            if(sta&8)cc--,b++;
            if(b&1)sum=(sum-c[r*cc][k]+mod)%mod;
            else sum=(sum+c[r*cc][k])%mod;
        }
        cout<<"Case "<<cas<<": "<<sum<<endl;
    }
    return 0;
}

递推关系

Ingenuous CubrencyUVA - 11137 (背包)

题意

求将n写成诺干个正整数的立方和有多少种方案

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod=1e6+7;
const int maxn=1e6+10;
const ll inf=0x3f3f3f3f3f3f3f3fLL;
ll dp[maxn];
int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    std::cout.tie(0);
    dp[0]=1;
    for(int i=1;i<=21;i++){
        for(int j=0;j<=10000;j++){
            dp[i*i*i+j]+=dp[j];
        }
    }
    int n;
    while(cin>>n)cout<<dp[n]<<endl;
    return 0;
}

Exploring PyramidsUVALive - 3516 (记忆化搜索)

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod=1e9;
const int maxn=1e6+10;
const ll inf=0x3f3f3f3f3f3f3f3fLL;
char s[maxn];
ll dp[400][400];

ll dfs(int i,int j){
    if(i==j)return 1;
    if(s[i]!=s[j])return 0;
    ll &ans=dp[i][j];
    if(ans>=0)return ans;ans=0;
    for(int k=i+2;k<=j;k++)
        if(s[i]==s[k])
        ans=(ans+1LL*dfs(i+1,k-1)*1LL*dfs(k,j))%mod;
    return ans;
}

int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    std::cout.tie(0);

    while(cin>>s){
        memset(dp,-1,sizeof(dp));
        cout<<dfs(0,strlen(s)-1)<<endl;
    }
    return 0;
}

Investigating Div-Sum PropertyUVA - 11361 （数位DP）

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod=1e9;
const int maxn=1e6+10;
const ll inf=0x3f3f3f3f3f3f3f3fLL;
ll dp[30][150][150];
int nu[30];
int k;
ll dfs(int pos,int num,int sum,bool limit){
    if(pos==-1)return (num%k==0)&&(sum%k==0);
    if(!limit&&dp[pos][num][sum]!=-1)return dp[pos][num][sum];
    int up=limit?nu[pos]:9;

    int ans=0;
    for(int i=0;i<=up;i++){
        ans+=dfs(pos-1,(num+i)%k,(sum*10+i)%k,limit&&i==nu[pos]);
    }
    if(!limit)dp[pos][num][sum]=ans;
    return ans;
}
ll ac(ll x){
    ll pos=0;
    while(x){
        nu[pos++]=x%10;
        x/=10;
    }
    return dfs(pos-1,0,0,true);
}
int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    std::cout.tie(0);
    int t;
    cin>>t;
    while(t--){
        ll a,b;
        cin>>a>>b>>k;
        if(k>90){
            cout<<0<<endl;continue;
        }
        memset(dp,-1,sizeof(dp));
        cout<<ac(b)-ac(a-1)<<endl;
    }
    return 0;
}
/*
999
1 1000 4

*/

数论

Always an integer UVALive - 4119 （欧拉函数）

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod=1e9;
const int maxn=4e6+10;
const ll inf=0x3f3f3f3f3f3f3f3fLL;

const int N = 4e6+10;
int phi[N] = {0, 1};
void caleuler()
{
    for (int i = 2; i < N; i++)
        if (!phi[i])
            for (int j = i; j < N; j += i)
            {
                if (!phi[j]) phi[j] = j;
                phi[j] = phi[j] / i * (i - 1);
            }
}
ll s[maxn],f[maxn];
void init(){
    caleuler();
    for(int i=1;i<N;i++){
        for(int j=i*2;j<N;j+=i)f[j]+=i*phi[j/i];
    }
    for(int i=2;i<N;i++)s[i]=s[i-1]+f[i];
}
int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    std::cout.tie(0);
    init();
    int n;
    while(cin>>n&&n){
        cout<<s[n]<<endl;
    }
    return 0;
}

Emoogle GridUVA - 11916 （大步小步算法）

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod=1e8+7;
const int maxn=550;
const ll inf=0x3f3f3f3f3f3f3f3fLL;

int n,m,k,b,r,x[maxn],y[maxn];
set<pair<int,int> >bset;

ll fastpow(ll a,ll n){
    ll ans=1;
    while(n){
        if(n&1)ans=(ans*a)%mod;
        a=(a*a)%mod;
        n>>=1;
    }
    return ans;
}

ll inv(ll a){
    return fastpow(a,mod-2);
}

ll mul_mod(ll a,ll b){
    return 1LL*a*b%mod;
}

int log_mod(int a,int b){
    int m,v,e=1,i;
    m=(int)sqrt(mod);
    v=inv(fastpow(a,m));
    map<int,int>x;
    x[1]=0;
    for(i=1;i<m;i++){
        e=mul_mod(e,a);
        if(!x.count(e))x[e]=i;
    }
    for(i=0;i<m;i++){
        if(x.count(b))return i*m+x[b];
        b=mul_mod(b,v);
    }
    return -1;
}

int count(){
    int c=0;
    for(int i=0;i<b;i++){
        if(x[i]!=m&&!bset.count(make_pair(x[i]+1,y[i])))c++;
    }
    c+=n;
    for(int i=0;i<b;i++)if(x[i]==1)c--;
    return mul_mod(fastpow(k,c),fastpow(k-1,1LL*n*m-b-c));
}

int doit(){
    int cnt=count();
    if(cnt==r)return m;

    int c=0;
    for(int i=0;i<b;i++)
        if(x[i]==m)c++;
    m++;
    cnt=mul_mod(cnt,fastpow(k,c));
    cnt=mul_mod(cnt,fastpow(k-1,n-c));
    if(cnt==r)return m;
    return log_mod(fastpow(k-1,n),mul_mod(r,inv(cnt)))+m;
}

int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    std::cout.tie(0);
    int t;
    cin>>t;
    for(int p=1;p<=t;p++){
        cin>>n>>k>>b>>r;
        bset.clear();
        m=1;
        for(int i=0;i<b;i++){
            cin>>x[i]>>y[i];
            if(x[i]>m)m=x[i];
            bset.insert(make_pair(x[i],y[i]));
        }
        cout<<"Case "<<p<<": "<<doit()<<endl;
    }
    return 0;
}

组合游戏

Playing With StonesUVALive - 5059 (sg函数)

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const ll mod=1e8+7;
const int maxn=550;
const ll inf=0x3f3f3f3f3f3f3f3fLL;
ll sg(ll x){
    return x%2==0?x/2:sg(x/2);
}
int main()
{
    std::ios::sync_with_stdio(false);
    std::cin.tie(0);
    std::cout.tie(0);
    int t;
    cin>>t;
    while(t--){
        ll n,a,v=0;
        cin>>n;
        for(int i=0;i<n;i++){
            cin>>a;
            v^=sg(a);
        }
        if(v)cout<<"YES
";
        else cout<<"NO
";
    }
    return 0;
}