Sigma (化简）牛客多校第一场 -- Integration

思路：

可以裂项化简，类似找规律，可以两项、三项代进去试试看。

 #define IOS ios_base::sync_with_stdio(0); cin.tie(0);
 #include <cstdio>//sprintf islower isupper
 #include <cstdlib>//malloc  exit strcat itoa system("cls")
 #include <iostream>//pair
 #include <fstream>
 #include <bitset>
 //#include <map>
 //#include<unordered_map>
 #include <vector>
 #include <stack>
 #include <set>
 #include <string.h>//strstr substr
 #include <string>
 #include <time.h>//srand(((unsigned)time(NULL))); Seed n=rand()%10 - 0~9;
 #include <cmath>
 #include <deque>
 #include <queue>//priority_queue<long long, vector<long long>, greater<long long> > q;//less
 #include <vector>//emplace_back
 //#include <math.h>
 //#include <windows.h>//reverse(a,a+len);// ~ ! ~ ! floor
 #include <algorithm>//sort + unique : sz=unique(b+1,b+n+1)-(b+1);+nth_element(first, nth, last, compare)
 using namespace std;//next_permutation(a+1,a+1+n);//prev_permutation
 #define mem(a,b) memset(a,b,sizeof(a))
 #define fo(a,b,c) for(a=b;a<=c;++a)//register long long i
 #define fr(a,b,c) for(a=b;a>=c;--a)
 #define pr printf
 #define sc scanf
 void swapp(long long &a,long long &b);
 double fabss(double a);
 long long maxx(long long a,long long b);
 long long minn(long long a,long long b);
 long long Del_bit_1(long long n);
 long long lowbit(long long n);
 long long abss(long long a);
 //const long long INF=(1LL<<60);
 const double E=2.718281828;
 const double PI=acos(-1.0);
 const long long inf=(<<);
 const double ESP=1e-;
 const long long mod=(long long)1e9+;
 const long long N=(long long)1e6+;

 long long a[N];
 long long qpow(long long a,long long b,long long mod)
 {
     long long ans;
     a%=mod;
     ans=;
     while(b!=)
     {
         if(b&)
             ans=(ans*a)%mod;
         b/=;
         a=(a*a)%mod;
     }
     return ans;
 }

 int main()
 {
     long long n,i;
     while(~sc("%lld",&n))
     {
         for(i=;i<=n;++i)
             sc("%lld",&a[i]);
         long long ans=;
         for(i=;i<=n;++i)
         {
             long long temp=a[i]*;
             for(long long j=;j<=n;++j)
             {
                 if(j==i)
                     continue;
                 temp*=((a[j]*a[j]-a[i]*a[i])%mod+mod)%mod;
                 temp%=mod;
             }
             ans+=qpow(temp,mod-,mod);
             ans%=mod;
         }
         pr("%lld\n",ans);
     }
     return ;
 }

 /**************************************************************************************/

 long long maxx(long long a,long long b)
 {
     return a>b?a:b;
 }

 void swapp(long long &a,long long &b)
 {
     a^=b^=a^=b;
 }

 long long lowbit(long long n)
 {
     return n&(-n);
 }

 long long Del_bit_1(long long n)
 {
     return n&(n-);
 }

 long long abss(long long a)
 {
     return a>?a:-a;
 }

 double fabss(double a)
 {
     return a>?a:-a;
 }

 long long minn(long long a,long long b)
 {
     return a<b?a:b;
 }